Mínimos locais de funcionais com dependência especial via Γ-convergência: com e sem vínculo

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(2) Γ.

(3) Γ.

(4) Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar. B586mL. Biesdorf, João. Mínimos locais de funcionais com dependência especial via Γ–convergência: com e sem vínculo / João Biesdorf. -São Carlos : UFSCar, 2011. 84p. Tese (Doutorado) -- Universidade Federal de São Carlos, 2011. 1. Cálculo das variações. 2. Gama convergência. 3. Solução estacionária estável. 4. Mínimos de funcionais. 5. Desigualdade isoperimétrica. 6. Equações de reação de difusão (Matemática) I. Título. CDD: 515.64 (20a).

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(7) 2. 3. Γ.

(8) 2. 3. Γ 2. 3.

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(10) ⎧ ⎪ ⎨. = 2 (a(x)∇u ) + f (x, u ), (t, x) ∈ R+ × Ω u (0, x) = φ(x), x ∈ Ω ⎪ ⎩ ∂u = 0, (t, x) ∈ R+ × ∂Ω ∂ν ∂u ∂t. >0. Ω ⊂ Rn (n ≥ 2) ∂Ω a ∈ C 1 (Ω) θ, g1 , g2 ∈ C 1 (Ω). C3 ν C 1 (Ω × R). f ∈. (f1 ) g1 (x) < θ(x) < g2 (x) f (x, ·) f (x, θ(x)) = f (x, g2 (x)) ≡ 0 ∂2 f (x, g1 (x)) < 0 ∂2 f (x, g2 (x)) < 0 (f2 ). g2 (x) g1 (x). f (x, ξ)dξ = 0. (f3 ) s (f4 ). . f (x, g1 (x)) = x∈Ω. c1 , c2 , s0 |s| ≥ s0. p≥2 c1 |s|p ≤ F (x, s) ≤ c2 |s|p v F (x, v) = − 12 g1 (x) f (x, s)ds. a(·)F (·, ·) ∈ C 1 . F (x, ·) g1 (x) g2 (x). F ≥ 0. x ∈ Ω,. n = 2, 3 g1 , g2 E : L (Ω) → [0, ∞] (1.1) . 2 1 a(x) |∇v| + F (x, v) dx, v ∈ H 1 (Ω) Ω E (v) := ∞, 1.

(11) F (·, ·) E. (1.1). E (1.1). Γ. (E )>0 {E }>0. Γ E0 (v) =. Ω. h(x) |Dχv=g2 | ,. →0. v ∈ BV (Ω; {g1 , g2 });. ∞. h(x) := 2. g2 (x). a(x)F (x, s)ds. g1 (x). F (·, ·) h(·) L1 (Ω). u0 {u }>0 →0. 0. E. u −u0 L1 (Ω) →. L1 (Ω). u0 ∈ L (Ω) 1. E0. E0. ρ>0 0 < u − u0 L1 (Ω) < ρ. E0 (u0 ) < E0 (u). E0. u0 g1. g2. g1 S. Ω1 , Ω 2 Ω1. g2. Ω. u0 u0. Ω. Ω2 a, f, g1 Ω. S. Ω. g2 S Ω u0. Γ. E0 E0 n = 2, 3,. α. β. g1 (x). g2 (x).

(12) S⊂Ω Λ : S × Iδ → R Λ(y, s) := h(y + sν(y)). n−1

(13). (1 − sκi (y)),. i=1. s=0. ν(y) κi (y) S y Iδ := (−δ, δ) x ∈ Sδ = {y + sν(y) : y ∈ S, s ∈ Iδ } s ∈ Iδ. x = y + sν(y). S⊂Ω Ωα. y∈S. S Ωβ E0. L1 (Ω). δ>0 y∈S. Ω u0 = αχΩα + βχΩβ ρ>0 u ∈ L1 (Ω). E0. a(·). f (·, s) h∗ (y, s). Ω s=0. y∈S. Sδ δ L1 (Sδ ). S L1 (Ω) L1 (Sδ ). n=2. L1 (Sδ ) n=2. n=2 u ∈ BV (Ω; {α, β}) L1 s ∈ ( 2δ , δ) u(·, s) Hn−2 = H0 A ∂∗ {u = α} ∩ ∂∗ {u = β} =

(14) ∅. n=2 S. S ∂∗ A. u H (∂∗ {u = α} ∩ ∂∗ {u = β}) ≥ 1, 0. h|Du(·, s)| = (β − α) S. hdH0 ∂∗ {u=α}∩∂∗ {u=β}. ≥ (β − α) inf {h(y + sν(y)}H0 (∂∗ {u = α} ∩ ∂∗ {u = β}) y∈S. ≥ (β − α) inf {h(y + sν(y)}. y∈S. S. u. h|Du(·, s)|.

(15) ρ > 0 ρ>0 u n > 2. n = 3 Hn−2. h|Du(·, ±s)| = (β − α) S. hdHn−2 ∂∗ {u=α}∩∂∗ {u=β}. ≥ (β − α) inf {h(y ± sν(y)}Hn−2 (∂∗ {u = α} ∩ ∂∗ {u = β}). y∈S. Hn−2 (∂∗ {u = α} ∩ ∂∗ {u = β}) ρ>0 n=3. n=2. δ>0 M ⊂R Hn (A) < ρ˜. M ⊂ R2 ρ˜ > 0. s ∈ ( 2δ , δ). A ⊂ M × ( 2δ , δ). Hn−1 ((M × {s}) ∩ A) ≤ Hn−1 (∂∗ A\(∂∗ A ∩ ∂∗ (M × [s, δ]).. Hn−1 (∂∗ A\(∂∗ A ∩ ∂∗ (M × [s, δ]) = Γ. E0. S. Sk M ρ˜k > 0 Sk ⊂ S. M ×[s, δ] (A).. (k = 1, . . . , k0 ) Mk ⊂ R Mk ρ˜ > 0 M Sk , n−1. Mk S Sδ Sk,δ ⊂ Sδ Sk,δ = {y + sν(y) : y ∈ Sk , s ∈ (−δ, δ) S y ∈ Sk Sk,δ Mk × (−δ, δ). ν(y). L1 (Sδ ) L1 (Sk,δ ). k (k = 1, . . . , k0 ) 0 < u − u0 L1 (Sk,δ ) < ρk. ρk > 0.

(16) E0 (u0 ) < E0 (u) L1 (Sk,δ ). ρ = min{ρk } 0 < u − u0 L1 (Sδ ) < ρ,. E0 (u0 ) < E0 (u) L1 (Sδ ) L1 (Sk,δ ) ρ˜k > 0 L1 (Sk,δ ). ρk. u0. h∗ (y, ·). s=0∀y∈S. (u )>0. (E )>0 u0 = αχΩα. α. β S. L1 (Ω) + βχΩβ u S. u0. u E. u. S L1 (Ω) Ω M × (−δ, δ) Ω. δ > 0. M ⊂ R. Ω :=. n−1. Ω. E. Ω. Γ. {v ∈ L (Ω) :. v dLn = m}. 1. Ω. Ω. m= Ω−. Ln (Ω) 2. (α+β). S. Ω+ Ω− = {(x, s) ∈ M × (−δ, 0)}. S := M ×{0}. E0.

(17) Ω+ = {(x, s) ∈ M × (0, δ)}. ∂S ⊂ ∂Ω, ∂S ∩ ∂Ω = ∅.. E Γ E0 u0 = αχΩ− + βχΩ+ Γ. E0. E0. {E }>0. → 0 u0. E0.

(18) [ ]. Ω ⊂ Rn. |Du| := sup u. u ∈ L1loc (Ω). |Du| (Ω) :=. . Ω. g=. n. g dx : g = (g1 , . . . , gn ) ∈. Cc1 (Ω. :R ). |g(x)| ≤ 1. n. |Du| (Ω) |u| dx < ∞ |Du| < ∞ Ω Ω BV. ∂gi i=1 ∂xi. u u(Ω) ⊂ B. Ω. Rn Ω ⊂ Rn. A A. x∈Ω ,. Ω. Ω Ω (A). := |DχA | (Ω).. Ω u ∈ BV (Ω; B)..

(19) Rn (A). P (A). Ω. (Ω). Ω Ω ⊂ Rn Ω. A Ω. A. Ω (A). := |DχA | (. (Ω)).. Rn. Hδn (A) = inf. ∞ i=1. α(n). diam(Ci ) 2. n. n. : A⊂. π2 , e Γ(n) = α(n) = Γ(n). ∞ . Ci , diam(Ci ) ≤ δ. i=1. ∞. e−x xn−1 dx.. 0. Hn (A) = lim Hδn (A). δ→0. [ ] Hn. Ln. Ln [ ]. (Ω)).. A ⊂ Rn C2 Ω = Rn , |DχA |(Rn ) = Hn−1 (∂A).. C2. |DχA |(Ω) = Hn−1 (∂A ∩.

(20) Ω = R2 A = {x ∈ R2 : x < 1} A = A\{(x1 , x2 ) ∈ R2 : ∂(A) A x2 = 0 e 0 < x1 < 1} 2 2 2 ∂A = {x ∈ R : x = 1} ∂A = {x ∈ R : x = 1} ∪ {(x1 , x2 ) ∈ R : x2 = 0 e 0 ≤ H1 (∂A ) = H1 (∂A) + 1 x1 < 1} L2 ({(x1 , x2 ) ∈ R2 : x2 = 0 e 0 < x1 < 1}) = 0 |DχA | (Ω) = |DχA | (Ω). A. A. E. E. x ∈ ∂∗ E,. E E lim sup. Ln (B(x, r) ∩ E) >0 rn. lim sup. Ln (B(x, r)\E) > 0. rn. r→0. r→0. |DχE | = Hn−1 ∂∗ E. Ω ⊂ Rn. |DχE |(Ω) = Hn−1 (∂∗ E ∩ Ω), E ⊂ Rn. x Ln (B(x, r) ∩ E) r→0 Ln (B(x, r)) lim. Ln (B(x, r) = α(n)rn 0 ∂∗ E ⊂ ∂E. α(n) E ∂E. 1. x. E E E. ∗. ∂ E x ∈ ∂ ∗E. ∗. ∂ E ⊂ ∂∗ E ⊂ ∂E Hn−1 (∂∗ E\∂ ∗ E) = 0..

(21) Ω ⊂ Rn P ∈ ∂Ω U = U (P ) ⊂ Rn. (x, s), x ∈ Rn−1 , s ∈ R, ϕP := ϕ : Rn−1 → R. |ϕ(x) − ϕ(y)| ≤ CP |x − y|. x, y ∈ Rn−1 , CP < ∞;. U ∩ Ω = {(x, s) : s > ϕ(x)} ∩ U.. [ ] f : Ω → Rm g : Ω → R, g|f −1 (y). Ln. g(x)J(f )(x) dx = Ω ⊂ Rn f |Du| =. ∞. g dH. |Du| =. |Du| = Ω. . dξ.. |Dχ{u(x)<t} | dt. Ω. . dH. −∞. n−1. Ω∩∂∗ {u>ξ}. ∞. ∞. Ω. f . −∞. Ω. dy.. f −1 (y). f dH. n−m. . −∞. Ω. y ∈ Rm. . u ∈ BV (Ω). Hn−m. Rm. Ω. n ≥ m, Lm. . Ω ⊂ Rn. n−1. dt.. Ω∩∂∗ {u>t}. f f. c1 (n) E ⊂ Rn. n ≥ 1 Hn (E). n−1 n. ≤ c1 (n)Hn−1 (∂∗ E).. n≥2 H (E) ≤ c1 (n) Rn |DχE | = c1 (n)Hn−1 (∂E) n≥2 n = 1 L(E) > 0 n. n−1 n. . c1 (1) =. 1 2.

(22) f : Ω ⊂ Rn → R m .. L ∈ Rm. f. y→x ap lim f (y) = L, y→x. > 0,. Ln (B(x, r) ∩ {|f − L| ≥ }) = 0. r→0 Ln (B(x, r)) lim. L = f (x). f. x (2.15). L f. Ln. Ln. f [ ]. t→a. x a x∈Ω f = O(g). f : Ω × R → R, f (x, t) = O(g(t)) k>0 |f (x, t)| ≤ k|g(t)|. {E }>0. {E }>0. {E }>0 : L1 (Ω) → [0, ∞] Γ. u ∈ L1 (Ω) lim inf →0 E (u ) ≥ E0 (u) u ∈ L1 (Ω) j→∞. vj −→ u. L1 (Ω). E0. {u } {vj }. →0. L1 (Ω) L1 (Ω). L1 (Ω). u.

(23) limj→∞ Ej (vj ) = E0 (u) E0. Γ. {E }>0. E0 = Γ− lim→0 E. v˜. H 1 (Ω). v˜. >0 v(x, t). φ ∈ H 1 (Ω), φ − v˜ H 1 (Ω) < δ t>0. v(·, t)− v˜ H 1 (Ω) < v˜ Γ E0. δ>0 v(x, 0) = φ limt→∞ v(x, t) = v˜(x). E0. E. Γ. {v }>0. E. E (v ) ≤ C < ∞. L1 (Ω).. (f3 ) L1 (Ω) BV (Ω). L1 (Ω). Ω ⊂ Rn E0 u0 0 > 0 u. {u }<0 L1 (Ω). u − u0 L1 (Ω) → 0. E → 0.. Γ L1 (Ω). E E0 {E }>0 E0.

(24) E0 u u. A ⊂ Rn. C2 . d(x) =. x∈A dist(x, ∂A), x ∈ Ac −dist(x, ∂A),. [ ] ⎧ ⎪ d ⎪ ⎪ ⎪ ⎪ ⎪ Ln ⎨ |∇d(x)| = 1; ⎪ ⎪ ⎪ δ > 0, ⎪ ⎪ ⎪ ⎩ (∂A)δ. 1;. d. (∂A)δ = {x ∈ Rn ; |d(x)| < δ}. δ ∂A.. C1. M∗. M. f : Rn → Rm [Df ] f |det([Df (x)]∗ ◦ [Df (x)])|, J(f )(x) = |det([Df (x)] ◦ [Df (x)]∗ )|, |∇d(x)| = 1 |∇d(x)| = 1. d d. 2.20. d. n ≤ m; n ≥ m. J(d) = 1. t → Hn−1 (d(x) = t). (∂A)δ [ ]. [ ]. d (∂A)δ 2.20, S = ∂A, y0 ∈ ∂A x0 = y0 + sν(y0 ) ν(y) −1 d (0) y s = d(x0 ).. x 0 ∈ Sδ. ν(y0 ), y0 = 0 V y0 y0 , ϕ ∈ C ∞ (Ty0 d−1 (0) ∩ Vy0 ) (x1 , . . . , xn−1 , ϕ(x1 , . . . , xn−1 )) ∈ d−1 (0) ∀ x = (x1 , . . . , xn−1 , xn ) ∈ Vy0 , ∇ϕ(y0 ) = 0,.

(25) y0 = πn (y0 ) πn (y) = πn (y1 , . . . , yn−1 , yn ) := (y1 , . . . , yn−1 ), y := πn (y) [D2 ϕ]y = (ϕxi xj (y ))(n−1)×(n−1) , κi (y0 ) (i = 1, . . . , n − 1) [D2 ϕ]y 0. ν, y0 , y0 , κ1 , . . . , κn−1 ⎛ ⎜ (Dν)y = ⎝ 0. −κ1 · · · 0. ···. 0. 0. ⎞. ⎟ −κn−1 0 ⎠ . 0 0. Ξ0 : ((Ty0 ∂A) ∩ Vy0 ) × Iδ → Rn Ξ0 (x , s) = (x , ϕ(x ) + sν(x , ϕ(x )), x (DΞ)(y ,s) 0. Iδ := (−δ, δ). ⎛ 1 − sκ1 · · · ⎜ =⎝ 0. J(Ξ0 )(y0 , s). =. 0. ⎞. ⎟ 1 − sκn−1 0 ⎠ , 0 1. ··· n−1

(26). 0. {1 − sκi ((y0 , ϕ(x0 )))} .. i=1. δ ((Ty0 S) ∩ Vy0 ) × Iδ Iδ } x = y(x) + d(x)ν(y(x)). J(Ξ0 ) Ξ0 {(x , ϕ(x )) + sν(x , ϕ(x )) : x ∈ (Ty0 S) ∩ Vy0 d(x) y(x) C1 x . s∈. S S Ξ :S × Iδ → Sδ (y, s) → y + sν(y). J(Ξ)(x, s) =. n−1

(27) i=1. (1 − sκi (y)) .. δ >0.

(28) C2 [ ]. C2. BΔA. A BΔA = (B c ∩ A) ∪ (Ac ∩ B).. Ω ⊂ Rn Ω. A⊂Ω 0 < L (A) < L (Ω). n. Ah C 2,. i)∂Ah. ii)Ln ((Ah ∩ Ω)ΔA) → 0 iii). Ω (Ah ). →. h → ∞,. Ω (A),. iv)Hn−1 (∂Ah ∩ ∂Ω) = 0, v)Ln (Ah ∩ Ω) = Ln (A). h. n. B.

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(30) Γ. E0. {E }>0. E Γ. E E (u) :=. Ω. a(x) |∇u|2 + 1 F (x, u) dx,. u ∈ H 1 (Ω). ∞,. Ω ⊂ Rn. F 1 F (x, v) = − 2. f ∈ C 1 (Ω × R). v. f (x, s)ds, g1 (x). θ, g1 , g2 ∈ C 1 (Ω). (f1 ) g1 (x) < θ(x) < g2 (x) f (x, ·) f (x, θ(x)) = f (x, g2 (x)) ≡ 0 ∂2 f (x, g1 (x)) < 0 ∂2 f (x, g2 (x)) < 0 (f2 ). g2 (x) g1 (x). f (x, ξ)dξ = 0. (f3 ) s (f4 ) g1 (x). . f (x, g1 (x)) = x∈Ω. c1 , c2 , s0 |s| ≥ s0. p≥2 c1 |s|p ≤ F (x, s) ≤ c2 |s|p v F (x, v) = − 12 g1 (x) f (x, s)ds. a(·)F (·, ·) ∈ C 1 .. F (x, ·) g2 (x). F ≥0 >0. x ∈ Ω, a ∈ C 1 (Ω).

(31) E : L1 (Ω) → R ∪ ∞, v ∈ L1 (Ω). . Γ− lim E (v) = →0. Ω. h(x) |Dχv=g2 | ,. v ∈ BV (Ω; {g1 , g2 });. ∞. g2 (x). h(x) := 2. a(x)F (x, s)ds.. g1 (x). Γ a=1. F (x, u) = W (u). g1. h. g2. a = 1 F (x, u) = (u − g1 (x))2 (u − g2 (x))2. g1. g2. Γ E0 := Γ− lim E . →0. v ∈ BV (Ω; {g1 , g2 }). v ∈ H 1 (Ω). v → v. L1 (Ω). {v∗ } v∗ ∈ H 1 (Ω) v∗ → v. g1 (x) ≤ v∗ (x) ≤ g2 (x), L1 (Ω). lim inf →0 E (v ) ≥ lim inf →0 E (v∗ ) ⎧ ⎪ ⎨ g1 (x), ∗ v (x) := v (x), ⎪ ⎩ g2 (x),. x ∈ {x : v (x) < g1 (x)}, x ∈ {x : g1 (x) ≤ v (x) ≤ g2 (x)}, x ∈ {x : v (x) > g2 (x)}..

(32) v(x) = gi (x)}; (i = 1, 2). Ω=. v → g i. L1 (Ωi ). 2 i=1. v∗. Ωi := {x ∈ Ω :. Ωi. i = 1, 2. v∗ → gi. L1 (Ωi ). i = 1, 2. i=1. ∗ |v (x) − g1 (x)|dx = |v∗ (x) − g1 (x)|dx Ω1 {x∈Ω1 :v (x)<g1 (x)}. |v∗ (x) − g1 (x)|dx + {x∈Ω1 :v (x)>g2 (x)}. |v∗ (x) − g1 (x)|dx + {x∈Ω1 :g1 (x)≤v (x)≤g2 (x)}. |g2 (x) − g1 (x)|dx = {x∈Ω1 :v (x)>g2 (x)}. |v∗ (x) − g1 (x)|dx. + {x∈Ω1 :g1 (x)≤v (x)≤g2 (x)}. v∗. {x∈Ω1 :v (x)>g2 (x)}. |g2 (x) − g1 (x)|dx → 0. → 0.. v → g 1 L1 (Ω1 ) δ > 0 0 (δ) > 0 < 0 (δ) Ln ({x ∈ Ω1 : |v (x) − g1 (x)| ≥ δ}) < δ. 0 < δ < inf x∈Ω (g2 (x) − g1 (x)) 0 (δ) > 0 < 0 (δ) Ln ({x ∈ Ω1 : v (x) ≥ g2 (x)}) ≤ Ln ({x ∈ Ω1 : |v (x) − g1 (x)| ≥ δ}) < δ.. |g2 (x) − g1 (x)|dx ≤ sup g2 (x) − g1 (x)Ln ({x ∈ Ω1 : v (x) ≥ g2 (x)}) {x∈Ω1 :v (x)>g2 (x)}. x∈Ω. < δ sup(g2 (x) − g1 (x)). x∈Ω. δ. {x∈Ω1 :v (x)>g2 (x)}. v∗ → g1. L1 (Ω1 ). |g2 (x) − g1 (x)|dx → 0 i=2. + v∗ = −(v − g2 )− + (g2 − g1 ) + g1 ,. → 0, i=1.

(33) f+ = f− =. f > 0, f ≤0. f, 0,. −f, f < 0, f ≥ 0. 0,. g1 , g2 ∈ C 1 (Ω) ⊂ H 1 (Ω) v ∈ H 1 (Ω) v∗ ∈ H 1 (Ω) 1 ∗ ∗ 2 ∗ a(x) |∇v | + F (x, v ) dx E (v ) = Ω . 1 ∗ 2 ∗ a(x) |∇v | + F (x, v ) dx = {x∈Ω:v (x)<g1 (x)}∪{x∈Ω:v (x)>g2 (x)} . 1 + a(x) |∇v∗ |2 + F (x, v∗ ) dx {x∈Ω:g1 (x)≤v (x)≤g2 (x)}. a(x) |∇v∗ |2 dx = {x∈Ω:v (x)<g1 (x)}∪{x∈Ω:v (x)>g2 (x)} . 1 2 + a(x) |∇v | + F (x, v ) dx {x∈Ω:g1 (x)≤v (x)≤g2 (x)} F (x, gi (x)) = 0. (. {x∈Ω:v (x)<g1 (x)}∪{x∈Ω:v (x)>g2 (x)}. v∗ = gi. v∗ ). i = 1, 2). a(x) |∇v∗ |2 dx → 0. .. → 0,. Ωi . E (v ) ≥. 1 a(x) |∇v | + F (x, v ) dx. 2. {x∈Ω:g1 (x)≤v (x)≤g2 (x)}. lim inf →0 E (v ) ≥ lim inf →0 E (v∗ ), [ ], Ω ⊂ Rn ∂f ∂f , . . . , ∂x : Ω × [a, b] → R. ∂x1 n C1 ∂ϕ (x) = ∂xi. f : Ω × [a, b] → R g : Ω → [a, b]. g(x) a. C 1.. ϕ(x) =. g(x) a. f (x, t)dt. ∂f ∂g (x, t)dt + (x)f (x, g(x)) ∂xi ∂xi. v ∈ BV (Ω; {g1 , g2 }), {vh }h∈N ⊂ H 1 (Ω) B := {σ ∈ C01 (Ω, Rn ); |σ(x)| ≤ 1 ∀x ∈ Ω}. g1 (x) ≤ vh (x) ≤ g2 (x). vh → v. L1 (Ω).

(34) a. 3.1.. F . lim. h→∞. sup − σ∈B. Ω. = sup − σ∈B. v(x). vh (x). σ+ <. a(x)F (x, s)ds. ! " ∇x a(x)F (x, s)ds, σ > dx. g1 (x). ! ∇x a(x)F (x, s)ds, σ > dx .. v(x). σ+ <. g1 (x). g1 (x). a(x)F (x, s)ds g1 (x). t. ˜ t) := φ(x,. σ∈B. . t. a(x)F (x, s)ds. g1 (x). Ω. vh (x). + < g1 (x) ∇x a(x)F (x, s)ds, σ > . ˜ t2 ) − φ(x, ˜ t1 )| ≤ L|t2 − t1 | |φ(x,. σ. (f4 ). t1 , t2 ∈ [g1 (x), g2 (x)]. L>0. x∈Ω. . ˜ ˜ φ(x, vh )dx − φ(x, v)dx Ω Ω. ˜ ˜ |φ(x, vh ) − φ(x, v)|dx ≤ L|vh − v|dx. ≤ Ω. σ∈B. vh (x). lim −. h→∞. Ω. v(x). =− Ω. a(x)F (x, s)ds. a(x)F (x, s)ds. vh (x). σ+ <. g1 (x). Ω. g1 (x). v(x). σ+ <. g1 (x). ! ∇x a(x)F (x, s)ds, σ > dx. ! ∇x a(x)F (x, s)ds, σ > dx.. g1 (x). B v, g1 , g2 , B, a. . g2 (x). F. a(x)F (x, s)ds · σ(x). g1 (x). g2 (x). =. a(x)F (x, s)ds. 3.4.. ". g1 (x). . =. =. g2 (x). a(x)F (x, s)ds · σ(x). g1 (x) g2 (x) g1 (x). a(x)F (x, s)ds a(x)F (x, s)ds. ∇x. a(x)F (x, s)ds, σ >. g1 (x). ". g1 (x) g2 (x). g2 (x). σ+ <. σ+ < ∇x. g1 (x) g2 (x). ∇x. σ+ < . g2 (x). a(x)F (x, s)ds, σ > a(x)F (x, s)ds, σ >. g1 (x). a(x)F (x, gi (x))∇vh (x) = 0 (i = 1, 2))..

(35) B, a. < Ω. 3.4. F. σ∈B. a(x)F (x, vh )∇vh , σ > dx. =− Ω. vh (x). a(x)F (x, s)ds. g1 (x). vh (x). σ+ <. ! ∇x a(x)F (x, s)ds, σ > .dx. g1 (x). φ(x, t) = a(x)F (x, g1 (x))∇vh (x) = 0,. ∇φ(x, vh ) =. vh ∈ H 1 (Ω). vh (x). ∇x. t g1 (x). a(x)F (x, s)ds. a(x)F (x, s)ds + a(x)F (x, vh (x))∇vh (x),. g1 (x). 3.1. a F g1 g 2. θ ∈ C 1 (Ω). g1 (x) ≤ θ(x) ≤ g2 (x) . = (a)−1 (x)F (x, s) Z(x, 0) = θ(x) dZ ds. Ω×R g1 (x) < Z(x, s) < g2 (x);. s ∈ R; |∇x Z(x, s)| ∈ L∞ (Ω × R). k1 , k2 , a, b. g2 (x) − Z(s, x) ≤ k1 e−as. s −s. Z(s, x) ≤ g1 (x)k2 ebs. lim Z(x, s) = g2 (x);. s→∞. lim Z(x, s) = g1 (x).. s→−∞. v ∈ L1 (Ω) lim inf →0 E (vj ) ≥ E0 (v). {vj }. L1 (Ω). v.

(36) {vj }. L1 (Ω) v. limj→∞ Ej (vj ) ≤. E0 (v) v ∈ L1 (Ω) v → v L1 (Ω). {v }>0 lim inf →0 E (v ) lim inf →0 E (v ). (vh )h∈N. v h = vh lim Eh (vh ) = lim inf E (v ) ∈ R. →0. h→∞. F (x, vh )dx ≤ Ω. 2h a(x) |∇vh |2 + F (x, vh ) dx. Ω. = h Eh (vh ),. h Eh (vh ) → 0. h → ∞.. F (x, v)dx = 0. v(x) ∈ {g1 (x), g2 (x)}q.t.p. x ∈ Ω.. Ω. g1 (x) ≤ vh (x) ≤ g2 (x). a2 + b2 ≥ 2ab. a, b ∈ R. B. 3.4 1 h a(x) |∇vh | + F (x, vh ) dx Eh (vh ) = h Ω 1 2 2 h ( a(x) |∇vh |) + ( F (x, vh )) dx = h Ω. a2 + b2 ≥ 2ab) ≥ 2 |∇vh | a(x)F (x, vh )dx ( Ω ≥ sup 2 a(x)F (x, vh ) < ∇vh , σ > dx σ∈B Ω vh (x) a(x)F (x, s)ds σ = sup −2 . 2. σ∈B. +<. Ω. vh (x) g1 (x). h→∞. g1 (x). ! ∇x a(x)F (x, s)ds, σ > dx. 3.6).. φ.

(37) lim Eh (vh ) ≥ lim. h→∞. +<. sup −2. h→∞. σ∈B. vh (x) g1 (x). . Ω. Ω. v(x). +< . v(x). a(x)F (x, s)ds. σ. g1 (x). = sup −2 σ∈B. +<. σ. g1 (x). ! ∇x a(x)F (x, s)ds, σ > dx. g1 (x). a(x)F (x, s)ds. ! " ∇x a(x)F (x, s)ds, σ > dx. = sup −2 σ∈B. vh (x). χv=g2 (x) Ω. g2 (x) g1 (x). . . a(x)F (x, s)ds. σ. g1 (x). ! ∇x a(x)F (x, s)ds, σ > dx . χv=g2 (x). = sup −2 σ∈B. g2 (x). 3.4). Ω. g2 (x). v ∈ BV (Ω; {g1 , g2 })). " a(x)F (x, s)ds · σ(x) dx. g1 (x). 3.5). . g2 (x) a(x)F (x, s)ds|Dχv=g2 | =2 Ω g1 (x). h(x)|Dχv=g2 | = E0 (v) = Ω. (3.8) {i }i∈N. i → 0. L1 (Ω). v ∈ L1 (Ω),. ρ i → v lim Ei (ρi ) ≤ E0 (v).. i→∞. v∈ / BV (Ω; {g1 (x), g2 (x)}), ∞. E0 (v), E0 (v) =. (3.9). Z. v ∈ BV (Ω; {g1 (x), g2 (x)}) ∂A C 2 − d A (3.6). A = {x ∈ Ω; v(x) = g2 (x)}. 2.20. ⎧ ⎪ g2 (x), ⎪ ⎪ ⎪ √ (s−2√) ⎪ ⎪ [g (x) − Z (x, 1/ )] √ + g2 (x), ⎪ 2 ⎨ g (x, s) := Z (x, s/) , ⎪ √ ⎪ √ ) (s+2 ⎪ ⎪ √ [Z (x, −1/ ) − g1 (x)] + g1 (x), ⎪ ⎪ ⎪ ⎩ g1 (x),. √ s > 2 ; √ √ ≤ s ≤ 2 ; √ |s| < ; √ √ − 2 ≤ s ≤ − ; √ s < −2 ..

(38) ρ (x) := g (x, d(x)). v(x) = g1 (x)χΩ\A + g2 (x)χA (x),. |ρ (x) − v(x)|dx =. L1 (Ω). ρ i → v. √ |d|<2 . Ω. (3.9). |ρ − v|dx.. √ < d(x) < 2 x ∈ A,. √ √ 1 √ |ρ − v|dx = |(g (x) − Z(x, 1/ ))(d(x) − 2 |dx 2 √ √ √<d<2√ <d<2 . √ √ 1 |(g (x) − Z(x, 1/ ))(d(x) − 2 |J(d)(x)dx, =√ √ 2 <d<2√ √ 2.21, J(d)(x) = |∇d(x)|2 = 1, x |d(x)| < 2 , (2.12) m = 1 f = d, g(x) = √ √ |(g2 (x) − Z(x, 1/ ))(d(x) − 2 |, (3.11). 2√ √ √ 1 √ |ρ − v|dx = |t − 2 | |g (x) − Z(x, 1/ )|dHn−1 dt. 2 √ √ √ <d<2 d−1 (t) √ √ A {x ∈ Ω : < d(x) < 2 } √ √ Hn−1 (d−1 (t)) ≤ k k<∞ <t<2 √ 3.7 |g2 (x) − Z(x, 1/ )| < g2 (x) √. x. 1 √ |ρ − v|dx ≤ k sup g2 (x) √ √ x∈Ω <d<2 . √ 2 √. √ |t − 2 |dt. . 2√ √ (2 − t)2 1 = √ k sup g2 (x) − √ 2 x∈Ω 1 1 k = sup g2 (x) 2 = O( 2 ) → 0. 2 x∈Ω. . 1. |ρ − v|dx ≤ O( 2 ).. √ √ −2 <d< . (3.10), (3.10) (3.10). |ρ (x) − v(x)|dx = Ω. {x ∈ Ω : 0 < d(x) <. √ |d|< . √. √ |ρ (x) − v(x)|dx + O( ). }. √ 0<d< . |ρ − v|dx =. √ 0<d< . [g1 (x) − Z(x, d(x)/)] dx. →0.

(39) Z(x, d(x)/). 2.12 s = t/. √. √ 1/ . |ρ − v|dx = 0. 0<d< . ≤. (3.16). m = 1 f = d g(x) = g2 (x)−. |g2 (x) − Z(x, s)|dHn−1 ds d=s. √ 1 sup√ Hn−1 (d(x) = t) √ sup(g2 (x) − g1 (x)) = O( ) x∈Ω {t:0≤t≤ }. √. √ |ρ − v|dx = O( ). → 0.. → 0.. 0<d< . √ L1 (Ω) ρ → v O( ) → 0. ρ (3.9) 1 2 a(x) |∇ρ | + F (x, ρ ) dx E (ρ ) = Ω . 1 2 a(x) |∇ρ | + F (x, ρ ) dx = √ |d|>2 . 1 2 a(x) |∇ρ | + F (x, ρ ) dx + √ |d|≤2 . (3.15) (3.17). (3.18). . 1 a(x) |∇ρ | + F (x, ρ ) dx √ |d|>2 . 1 2 a(x) |∇g2 (x)| + F (x, g2 (x)) dx = √ |d|>2 . a(x) |∇g2 (x)|2 dx = O() → 0. = √. 2. |d|>2 . (3.19). (3.20),. E (ρ ) =. √ |d|≤2 . . 1 a(x) |∇ρ | + F (x, ρ ) dx + O() h 2. → 0.. √ √ {x ∈ Ω : < d(x) < 2 }, √ 1 1 d(x) − 2 ∇d(x) √ g2 (x) − Z(x, √ ) + ∇g2 (x) ∇ρ (x) = ∇g2 (x) − ∇x Z(x, √ ) + √ $ # √ √ d(x)−2 d(x)−2 √ √1 ) √ (x) − ∇ Z(x, ∇g ≤ 1 2 x $ # ∇d(x) 1 √ √ →0 g2 (x) − Z(x, ) .

(40) →0. . ∇g2 (x). 3. √. . √. a(x) |∇ρ |2 dx = O( 2 ). → 0.. <d(x)<2 . . 3 1 1 2) F (x, ρ a(x) |∇ρ | + F (x, ρ ) dx = √ )dx + O( √ <d(x)<2 2. √. √ <d(x)<2 . → 0.. 2.12 m = 1 f (x) = d(x) √. % √ & (d(x) − 2 ) 1 √ g(x) = F x, g2 (x) − Z x, 1/ + g2 (x) (3.23). 1 F (x, ρ )dx √ √ <d(x)<2 √ 2√ . % √ & (d(x) − 2 ) 1 √ F x, g2 (x) − Z x, 1/ = √ + g2 (x) dHn−1 dt. d(x)=t 3.7,. k1. k := k1 sup{∂2 F (x, s) : x ∈ Ω. √. g2 (x) ≤ s ≤ g2 (x) + k1 e−a/ }. k˜ := k (n−1) sup√≤t≤2√ Hn−1 ({x : d(x) = t}). (3.24). 1 F (x, ρ )dx <d(x)<2 2√ ( 1 ' # −a/√ $ F x, k1 e ≤ √ 1 + g2 (x) dHn−1 dt d(x)=t 2√ √ # √ $ 1 −a/ −a/ sup{∂2 F (x, s) : x ∈ Ω g2 (x) ≤ s ≤ g2 (x) + k1 e } k1 e ≤ √ dHn−1 dt d(x)=t 2√ √ √ 1 # −a/√ $ ˜ −(n−1)a/ ke = √ dHn−1 dt = −(n−1) ke d(x)=t √. = O(m ). √. →0 (3.25). m ∈ N.. (3.23) . 3 1 a(x) |∇ρ | + F (x, ρ ) dx = O( 2 ) 2. √. √ <d(x)<2 . →0.

(41) . 3 1 a(x) |∇ρ | + F (x, ρ ) dx = O( 2 ) 2. √ √ −2 <d(x)<− . (3.21), (3.26) E (ρ ) =. (3.27), . 1 a(x) |∇ρ | + F (x, ρ ) dx + O() 2. √ |d|≤ . |d(x)| < ∂2 Z(x, t) :=. → 0.. √. → 0.. ρ (x) = Z(x, d(x)/).. ,. d Z(x, t), dt. F (Z(x, t)) ∂2 Z(x, t) = a(x). √. a∂2 Z =. √. F.. . 2 1 F (x, ρ (x)) + a(x) ∇ρ(x) dx √ |d(x)|< 2 !. 1 1 F (x, ρ (x)) + a(x) ∇x Z(x, d(x)/) + ∇d(x)∂2 Z (x, d(x)/) dx = √ |d(x)|< 2 !. 1 1 F (x, ρ (x)) + a(x)∇x Z(x, d(x)/) + ∇d(x) F (x, ρ (x)) dx = √ |d(x)|< . 1 F (x, ρ (x)) + a(x)|∇x Z(x, d(x)/)|2 = √ |d(x)|< 1 + 2a(x)∇x Z(x, d(x)/)∇d(x) F (x, ρ (x)) + F (x, ρ (x)) dx . . =. √ |d|< . √ 2 2 F (x, ρ ) + a|∇x Z| + 2 a∇x Z∇d F (x, ρ ) dx. . a|∇x Z|2 + 2a∇x Z∇d. # √ |d(x)|< . (3.31). F (ρ ). {x ∈ Ω : |d(x)| <. √. $ √ a|∇x Z|2 + 2 a∇x Z∇d F (x, ρ ) dx = O(1/2 ). (3.30),. (3.28),. E (ρ ) =. √ |d|≤ . √ 2 F (x, ρ )dx + O( ) . → 0.. }. → 0. → 0..

(42) 2.12 2 . s = t ,. F (x, ρ (x))|∇d(x)|dx. 2 F (x, Z(x, t/)dHn−1 dt = |t|<√ d(x)=t. =2 F (x, Z(x, s)dHn−1 ds. √ √ |d(x)|< . |s|< . d(x)=s. 2.23 √ x : d(x) < (2.7). δ, ν, κi s = t/ y = x − sν(y). J(y). (x,s). =. n−1

(43). √ 2 <δ x = x(y, s) = y + sν(y). t = d(x). (1 − sκi ).. i=1. (3.33). 2. |s|< √1. F (x(y, s), Z(x(y, s), s) d(y)=0. n−1

(44). (1 − sκi )dHn−1 ds.. i=1. 3.3, ∂ ∂s. Z(x(y,s),s). g1 (x(y,s)) Z(x(y,s),s). a(x(y, s)) F (x(y, s), t)dt. ( ∂ ' a(x(y, s)) F (x(y, s), t) dt ∂s g1 (x(y,s)) ∂ + a(x(y, s)) F (x(y, s), Z(x(y, s), s)) Z(x(y, s), s) ∂s Z(x(y,s),s) ( ∂ ' = a(x(y, s)) F (x(y, s), t) dt ∂s g1 (x(y,s)) + a(x(y, s)) F (x(y, s), Z(x(y, s), s))∇x Z(x(y, s), s)ν(y) + a(x(y, s)) F (x(y, s), Z(x(y, s), s))∂2 Z(x(y, s), s). =. (3.6), F (x(y, s), Z(x(y, s), s)) = (3.35),. a(x(y, s)) F (x(y, s), Z(x(y, s), s))∂2 Z(x(y, s), s).. Z(x(y,s),s) ∂ F (x(y, s), Z(x(y, s), s)) = a(x(y, s)) F (x(y, s), t)dt ∂s g1 (x(y,s)) ∂ − a(x(y, s)) F (x(y, s), Z(x(y, s), s)) Z(x(y, s), s) ∂ν Z(x(y,s),s) ' ( ∂ − (a(x(y, s)) F (x(y, s), t) dt. ∂s g1 (x(y,s)).

(45) (3.36). I 1 , I2 , I3. 2. |s|< √1. F (Z(x(y, s), s) d(y)=0. =2. +2. +2. |s|< √1. n−1

(46). I1 d(y)=0. (1 − sκi )dHn−1 ds. i=1. I2. n−1

(47). d(y)=0. (1 − sκi )dHn−1 ds. i=1. |s|< √1. (1 − sκi )dHn−1 ds. i=1. |s|< √1. n−1

(48). I3. n−1

(49). d(y)=0. (1 − sκi )dHn−1 ds.. i=1. ∂ a(x(y, s)) F (x(y, s), Z(x(y, s), s)) ∂ν Z(x(y, s), s). I2. |s|< √1. |s|< √1. I3 d(y)=0. =. =. =. |s|< √1. |s|< √1. |s|< √1. √ = O( ). n−1

(50). n−1

(51). Z(x(y,s),s). (1 − sκi ). g1 (x(y,s)). n−1

(52). ( ∂ ' a(x(y, s)) F (x(y, s), t) dt dHn−1 ds ∂s. Z(x(y,s),s). (1 − sκi ). ν(y)∇x. '. ( a(x(y, s)) F (x(y, s), t) dt dHn−1 ds. g1 (x(y,s)). d(y)=0 i=1. . →0. i=1. d(y)=0 i=1. d(y)=0. √ (1 − sκi )dHn−1 = O( ). (1 − sκi )dHn−1 ds. i=1. n−1

(53). d(y)=0. n−1

(54). Z(x(y,s),s). (1 − sκi ) g1 (x(y,s)). i=1. ( ∂ ' a(x) F (x, t) x=x(y,s) dt dHn−1 ds ∂ν. √ |∇x aF |L∞ (Ω×(inf x∈Ω g1 (x),supx∈Ω g2 (x))) < ∞.. → 0,. (3.32), (3.33), (3.34), (3.37), (3.38). E (ρ ) = 2. |s|< √1. I1 d(y)=0. n−1

(55) i=1. (3.39). √ (1 − sκi )dHn−1 ds + O( ). → 0..

(56) . I1. |s|< √1. n−1

(57). d(y)=0. (1 − sκi )dHn−1 ds. i=1. Z(x(y,s),s) ∂ (1 − sκi ) a(x(y, s)) F (x(y, s), t)dt dHn−1 ds = ∂s g1 (x(y,s)) |s|< √1 d(y)=0 i=1 . n−1 ∂

(58) = (1 − sκi ) |s|< √1 d(y)=0 ∂s i=1 "! Z(x(y,s),s) a(x(y, s)) F (x(y, s), t)dt dHn−1 ds. n−1

(59). g1 (x(y,s)). −. |s|< √1. d(y)=0. Z(x(y,s),s). ⎛. ⎡. ⎢ ⎣. n−1 . ⎞. ⎜ ⎝−λj. j=1. n−1

(60). ⎟ (1 − sκi )⎠. i=1 i

(61) =j. ! a(x(y, s)) F (x(y, s), t)dt dHn−1 ds. g1 (x(y,s)). II. (3.41). III √ |III | = O( ). ⎡ ⎢ ⎣. ⎛ n−1 . ⎞. ⎜ ⎝−λj. j=1. n−1

(62). ⎟ (1 − sκi )⎠. ⎤. i=1 i

(63) =j. → 0,. Z(x(y,s),s). . a(x(y, s)). g1 (x(y,s)). ⎥ F (x(y, s), t)dt⎦. → 0.. II =. |s|< √1. ∂ ∂s. n−1

(64). d(y)=0. Z(x(y,s),s). (1 − sκi ). . a(x(y, s)) F ((x(y, s), t)dt. "! dHn−1 ds. g1 (x(y,s)). i=1. d(y)=0. s= √1 ! (1 − sκi ) a(x(y, s)) F ((x(y, s), t)dt dHn−1 g1 (x(y,s)) i=1 s=− √1 ! Z(x(y, √1 ), √1 ) / n−1

(65) √ √ / √ (1 − κi ) a(x(y, 1/ )) F (x(y, 1/ ), t)dt dHn−1. d(y)=0. i=1. n−1

(66). =. =. n−1

(67). − d(y)=0. i=1. (1 +. Z(x(y,s),s). g1 (x(y, √1 )). √. κi ). Z(x(y,− √1 ),− √1 ) g1 (x(y,− √1 )). ! / √ / √ a(x(y, −1/ )) F (x(y, −1/ ), t)dt dHn−1.

(68) (3.40), (3.41), (3.42), (3.43). n−1

(69). E (ρ ) = 2 d(y)=0. d(y)=0. κi ). i=1 n−1

(70). −2. (1 −. √. (1 +. √. κi ). i=1. √ + O( ). Z(x(y, √1 ), √1 ) g1 (x(y, √1 )). Z(x(y,− √1 ),− √1 ) g1 (x(y,− √1 )). (3.44). (3.45). lim E (ρ ) =. 2. ! / √ / √ a(x(y, −1/ )) F (x(y, −1/ ), t)dt dHn−1. →0. ⎧ 0 √ lim→0 n−1 (1 − κi ) ⎪ i=1 ⎪ ⎪ 0n−1 √ ⎪ ⎪ (1 + κi ) lim ⎪ →0 i=1 ⎨ lim→0 x(y, √1 ) ⎪ ⎪ ⎪ lim→0 Z(x(y, √1 ), √1 ) ⎪ ⎪ ⎪ ⎩ lim→0 Z(x(y, − √1 ), − √1 ). →o. ! / √ / √ a(x(y, 1/ )) F (x(y, 1/ ), t)dt dHn−1. =. .. = g2 (y). 3.7). = g1 (y). 3.7). 2.1. g2 (y). ! a(y)F (y, t)dt dHn−1. g1 (y). d(y)=0. =1 =1 =y. h(y)dH. n−1. d(y)=0. =. h(y)dH. n−1. =. h(y)|DχA | = Ω. ∂A. Ω. h(y)|Dχ{v=g2 } |. = E0 (v). ∂A. C 2− A. A. m ∈ N, 1 χ A − χA 1 < j1 (m) j ≥ j1 (m) j 2m L (Ω) E0 (χA ) − E0 (χA ) <. h L∞ (Ω) < ∞, m ∈ N, j2 (m) j 1 j ≥ j (m) j(m) = max{j (m), j (m)}. 2 1 2 2m (3.46) Aj ρ,j → χAj L1 (Ω) Aj m ∈ N 1 (m) 1 |ρ,j(m) − χAj(m) | < 2m ≤ 1 (m) m∈N 2 (m) 1 |E (ρ,j(m) ) − E0 (χAj(m) )| < 2m ≤ 2 (m) (m) = min{1 (m), 2 (m)} Aj. vm = ρ(m),j(m) .. L1 (Ω). vm → χA {um }m∈N (3.9) |vm − χA |L1 (Ω) = ρ(m),j(m) − χA L1 (Ω) 1 ≤ ρ(m),j(m) − χAj(m) + χAj(m) − χA < , m L1 (Ω) L1 (Ω) E(m) (vm ) − E0 (χA ) = E(m) ( ρ(m),j(m) ) − E0 (χA ).

(71) 1 ≤ E(m) ( ρ(m),j(m) ) − E0 (χAj(m) ) + E0 (χAj(m) ) − E0 (χA ) < . m 3.1. A.

(72)

(73) u0 ∈ BV (Ω; {α, β}) E0. Γ. r > 0 Ω = M × (−δ, δ). δ >0. M ⊂ R2 C1. n=3. M ⊂R. n=2. . ρ >0 A+ ⊂ M ×( 2δ , δ) Hn (A+ ) < ρ Hn−1 ((M ×{l})∩A+ ) ≤ 1r 1 n−1 H (M ) 4. Ω (A. )<∞ l ∈ ( 2δ , δ) + Hn−1 ((M ×{l})∩A+ ) ≤ M ×(l,δ) (A ) +. − A− ⊂ M × (−δ, − 2δ ) Hn (A− ) < ρ Ω (A ) < ∞ − Hn−1 ((M × {l }) ∩ A− ) ≤ 1r l ∈ (−δ, − 2δ ) M ×(−δ,l ) (A ) Hn−1 ((M × {l }) ∩ A− ) ≤ 14 Hn−1 (M ). r > 0 M E0. u0 ∈ BV (Ω; {α, β}) δ > 0. u0.

(74) (2.4) u=α. u ∈ BV (Ω; {α, β}) A+ A− u0 = β u=β u0 = α 1 L u0. ρ > 0 M × ( 2δ , δ) M × (−δ, − 2δ ) n = 2 n = 3. n>3. n>3. n = 2, 3. n≥4. n = 2 n = 3. n = 2. u0. E0 2. L ⊂ Rn−1 × Rn−1 f : (L × L)\{(P, P ) : P ∈ N } → R+ P ∈N c>0. P. N ⊂ L lim inf (x,y)→(P,P ) f (x, y) = cP > 0 f ≥c. lim inf P ∈N UP f (x, y) ≥ (x, y) ∈ (L × L) ∩ UP (L × L)\ P ∈N UP (x0 , y0 ) ∈ (L × L)\ P ∈n UP 0 < f (x0 , y0 ) ≤ f (x, y) (x, y) ∈ (L × L)\ P ∈N UP c = min{f (x0 , y0 ), c2P : P ∈ N } cP 2. (2.1) (n − 1) Rn M ⊂ Rn−1 s. M.

(75) M × {s} 2. 3 θ>0. M E⊂M. 0<H. n−1. (E) <. 1 n−1 H (M ) 2. M (E). 0<θ≤. (E). .. (2.1) 0<θ≤ θ>0 E n=2 0 H (∂∗ E) = 2. Hn−2 (∂∗ E\∂∗ M ) . Hn−2 (∂∗ E). (4.1) θ>0. E (4.1) (4.1). H (·) H0 (∂∗ E\∂∗ M ) ≥ 1, n=2. θ = n=3. ∂∗ M. C. E. n−2. 1 2r. NM. 1. γ(P, Q) =. γ : (∂∗ M × ∂∗ M )\{(P , P ) : P ∈ NM } → R+ min{H1 (Ci (P, Q)) : Ci (P, Q) 0,. ∂∗ M \{P, Q}}. ∂∗ M \{P, Q}. P

(76) = Q γ σ : (∂∗ M × ∂∗ M )\{(P , P ) :. γ . P ∈ NM } → R. +. σ(P, Q) =. lim. inf. (P,Q)→(P ,P ). ∂∗ M. σ(P, Q) > 0. σ mσ > 0 σ ≥ mσ > 0. |P −Q| γ(P,Q)+|P −Q| 1 , 2. P =

(77) Q, P = Q,. P

(78) = Q, P = Q,. P ∈ NM . σ. (∂∗ M × ∂∗ M )\{(P , P ) : P ∈ NM }.

(79) c1 := c1 (2). mσ . . √ mσ 2 H2 (M )1/2 √ θ = min mσ , mσ 2 H2 (M )1/2 + 2c1 def. 1>θ>0 θ E⊂M 1 H (∂∗ E) ∩ ∂∗ M ) = 0. 1>θ 0 < H1 (∂∗ E ∩ ∂∗ M ) < (M ) P, Q ∈ ∂∗ E ∩ ∂∗ M ∂∗ M \{P, Q} P Q ∂∗ M \{P, Q} a, b c M (E). (E). =. (4.1) M (E). .. (M ) E⊂M. =. M (E). (E). (E). P Q ∂∗ E ∩ ∂∗ M \{P, Q}. ∂∗ E ∩ ∂∗ M. ∂∗ E ∩ ∂∗ M a≤b. M (E) H1 (∂∗ E. M (E). (4.3). =. P

(80) = Q. (∂∗ E) ∩ (∂∗ M ) |P − Q| ≤ M (E) b a ≤ b+c , c+a M (E). ≥. + ∩ ∂∗ M ) M (E) + γ(P, Q) |P − Q| ≥ = σ(P, Q) ≥ mσ ≥ θ. |P − Q| + γ(P, Q) H2 (E) < 12 H2 (M ) P. H2 (M \E) > M (E). =. ∂∗ E ∩ ∂∗ M (∂∗ E) ∩ (∂∗ M ) M \E. Q. ∂∗ M \{P, Q} (M \E) M ≥ mσ (M \E) 1 2 H (M ) 2 M (M \E). M (E). (E). ≥ mσ. (M \E) ≥ mσ. M (E). ≥. M (E). = H1 (∂∗ E ∩ ∂∗ M ) = mσ < 1 n=3. mσ. √. (M ). +. √. (M ). >. 1 2 1 1 H (M \E)1/2 > mσ √ H2 (M )1/2 . c1 c1 2. mσ c11 √12 H2 (M )1/2 mσ c11 √12 H2 (M )1/2 +. mσ 2 H2 (M )1/2 2 H2 (M )1/2 + 2c1. (M ). ≥ θ. M \E. (M ).

(81) n>3 P, Q (n − 3). ∂∗ M. n=4. σ (n − 3). ∂∗ M mσ > 0. σ (n − 1) n M M. θM. (n − 1) >0. lim inf M {θM } > 0 M. M. M = {(x, y) ∈ R2 : (x+1)2 +y 2 > 1; (x−1)2 +y 2 > 1 x2 +(y−1)2 < M (0, 0) En = {(x, y) ∈ R2 : (x+1)2 +y 2 > 1}, 1; (x − 1)2 + y 2 > 1 0 < y < n1 } M (En ). = H({(x, y) ∈ R2 : (x + 1)2 + y 2 > 1; (x − 1)2 + y 2 > 1 " 1 1 =2 1− 1− 2 , n. (En ) >. lim. 2 . n. / ' 2 1− 1− 2 n. n→∞. M (En ). (En ). <. 1 n2. ( =0. M (En ) . 2 n. y=. 1 }) n. (2.1)).

(82) lim. n→∞. M (En ). (En ). =0. θ>0. M. Hn−1 (E) < 12 Hn−1 (M ) 0 < m < 1. mHn−1 (M ). Hn−1 (E) < θ. M = {(x, y) ∈ R2 : |x| < y < 1} 1. 1 n. En = {(x, y) ∈ R2 : |x| < y <. < |x|} lim. n→∞. M (En ). (En ). =0. θ > 0 H. n−1. (E) <. 1 n−1 H (M ) 2. M. x, 0 < y <. 1 } x2. M = {(x, y) ∈ R2 : 1 < x, 0 < y < n∈N M (En ). (En ). 1 } x2. En = {(x, y) ∈ R2 : n <. = 0.. M θ>0. r > 0 Ω = M × (−δ, δ) ξ ∈ ( 2δ , δ). θ>0 η : [ δ20 , ξ] → [0, ∞). ⎧ ξ n−2 θ ⎪ n−1 ds η(s) η(t) = ⎪ c r t ⎪ 1 ⎪ ⎨ η(ξ) = 0 ⎪ η( 2δ ) ≤ 14 Hn−1 (M ) ⎪ ⎪ ⎪ ⎩ η>0 [ δ20 , ξ).. c1 := c1 (n − 1).

(83) η(t) =. '. θ (n−1)c1 r. (n−1. (ξ − t)n−1 (n = 2, 3). (4.8) ξ ∈ ( 2δ , δ) η( 2δ ) ≤ 14 Hn−1 (M ). δ 2. 1 n−1 H (M ) 4. θ c1 r. n = 2, n = 3. n≥2. r > 0 Ω = M × (−δ, δ) ρ > 0 δ 1 δ ψ(s)dH < ρ. l∈. 2. ( 2δ , δ)\. L I. θ ψ(l) < c1 r. c1 := c1 (n − 1). θ>0 ψ : [ 2δ , δ] → (0, Hn−1 (M )] I = {t ∈ [ 2δ , δ] : ψ(t) ≥ 14 Hn−1 (M )} n−2. ψ(s) n−1 ds. [l,δ]\I. ψ ψ(s) = H(n−1) (A+ ∩ (M × {s})). ψ c1. ψ n−2 n−1. θ I ρ > 0. Hn−1 (E) < 12 Hn−1 (M ) ρ ψ η δ l ∈ ( 2 , δ). ψ = η ψ η ξ ξ (δ − ξ) n−1 H (M )}, ρ = min{ η(t)dt, 4 δ/2. n = 2, 3.. ψ. I. . δ δ O = t ∈ , δ \ I : ∃s ∈ , t \ I 2 2 I˜ = I ∪ O. ϑ = L([0, δ]\ I). L([0, s]\ I) = L([0, t]\ I). ζ : [ 2δ , δ]\ I˜ → [ 2δ , ϑ] ˜ ζ(t) = L([0, t]\ I)..

(84) L(O) = 0. O. ζ L. ζ g : [ 2δ , ϑ] → (0, 14 Hn−1 (M )) g = ψ ◦ ζ −1 ψ. ϑ. ξ. g(s)ds < δ/2. ∃ l0 ∈ ( 2δ , ϑ). η(t)dt. δ/2. g(l0 ) <. θ c1 r. ϑ l0. n−2. g(s) n−1 ds. ϑ>ξ. ϑ. L(I) < δ − ξ. (δ − ξ) n−1 H (M ) 4 1 ≤ Hn−1 (M )L(I) 4. ρ ≤. <. ρ ). ψ(s)ds. δ. ψ(s)ds (. < δ/2. ≤ρ. I). I. δ I ⊂ [ , δ] ) 2. . ρ ). ϑ>ξ. δ def Δ = s ∈ [ , ξ) : g 2. s. g ∅ [δ/2, ξ). g. ξ = δ/2 ξ ξ ϑ g > δ/2 g > δ/2 η δ/2 Δ

(85) = ∅. [ 2δ , ξ]. g(s) ≤ η(s) . Δ

(86) =. Δ. g > η δ/2 < ξ. sup Δ = ξ. inf{g(s) : s ∈ [ 2δ , ξ] : g s} = 0. ϑ n−2 n−2 n−1 n−1 g > 0 g ds > 0 l0 ∈ [ 2δ , ξ] ξ l0 ϑ ϑ n−2 n−2 θ θ g(l0 ) ≤ g(s) n−1 ds ≤ g(s) n−1 ds. c1 r ξ c 1 r l0 sup Δ = ξ

(87) = ξ. ξ<ξ ϑ>ξ g>0 g. Δ l0 ∈ ( 2δ , ξ]. ξ ξ. ξ n−2 n−2 g(t) n−1 dt > ξ η(t) n−1 dt. [δ/2, ξ) δ/2 < ξ < ξ..

(88) • g. l0. • g(l0 ) ≤ η(l0 ); • •. ξ l0. ξ ξ. n−2. η(s) n−1 ds < n−2. g(t) n−1 dt >. ϑ ξ. ξ ξ. n−2. g(s) n−1 ds; n−2. η(t) n−1 dt. l0 = ξ˜. g. l0. ξ n−2 θ g(l0 ) ≤ η(l0 ) = η(s) n−1 ds c 1 r l0 ξ ξ n−2 n−2 θ θ n−1 = η(s) ds + η(s) n−1 (s)ds c 1 r l0 c1 r ξ ϑ ξ n−2 n−2 θ θ n−1 < g(s) ds + g(s) n−1 ds c1 r ξ c1 r ξ ϑ n−2 θ = g(s) n−1 ds c1 r ξ ϑ n−2 θ ≤ g(s) n−1 ds, c 1 r l0. l = ζ −1 (l0 ). ρ > 0 % & s ∈ 2δ , δ 0 . ψ : 2δ , δ → (0, Hn−1 (M )]. A+. A+ ψ. Ln−1. r > 0 δ > 0 Ω = M ×(−δ, δ) H(n−1) (A+ ∩(M ×{s})) =. s l (n−1) ψ(s) = H (A+ ∩(M ×{s})). Ln ψ. + Ω (A ). I. <∞.

(89) l∈ H. (n−1). (A ∩ (M × {l})) ≤ +. ≤ ≤ ≤ ≤ = l ∈. %δ. & , δ \I 2. %δ. & , δ \I 2. n−2 θ (H(n−1) (A+ ∩ (M × {s}))) n−1 dH(s) c1 r [l,δ]\I. θ H(n−2) (∂∗ (A+ ∩ (M × {s})))dH(s) r [l,δ]\I. 1 + (M ×{s} (A ∩ (M × {s})))dH(s) r [l,δ]\I. 1 + (M ×{s} (A ∩ (M × {s})))dH(s) r [l,δ] 1 (n−1) H (∂∗ (A+ ∩ (M × [l, δ]))\∂∗ (M × [l, δ])) r 1 + M ×(l,δ) (A ) r. 2.14) 4.3).

(90) E0 E0 (v) =. Ω. h(x) |Dχv=β | ,. v ∈ BV (Ω; {α, β});. ∞. β. h(x) := 2. a(x)F (x, s)ds.. α. >0 3 C ν f ∈ C 1 (Ω × R). Ω ⊂ Rn (n ≥ 2) ∂Ω a ∈ C 1 (Ω) θ ∈ C 1 (Ω) α, β. (f1 ) α < θ(x) < β f (x, ·) f (x, β) ≡ 0 ∂2 f (x, α) < 0 ∂2 f (x, β) < 0 (f2 ). β α. c1 , c2 , s0 |s| ≥ s0. s . x∈Ω. f (x, ξ)dξ = 0. (f3 ) (f4 ). f (x, α) = f (x, θ(x)) =. a(·)F (·, ·) ∈ C 1 . (f1 ) (f2 ) (f3 ). (f4 ). p≥2 c1 |s|p ≤ F (x, s) ≤ c2 |s|p v F (x, v) = − 12 g1 (x) f (x, s)ds.

(91) E0. u0 (i = 1, . . . , n − 1) ν(·). S⊂Ω y ∈S. κi (y) S. H(y). Λ:S×R→R Λ(y, s) := h(y + sν(y)). n−1

(92). (1 − sκi (y)). i=1. S. y ∈ S S. Λ Ω. s = 0 Ωα. Ωβ y∈S Λ(y, s). s=0 v0 (x) = αχΩα (x) + βχΩβ (x), x ∈ Ω E0. L1 H(y). S. y. y = (y , ϕ(y )) ∈ S, λy (s) := h(y + sν(y)).. λy (0) = (n − 1)h(y)H(y) ⎧ ⎪ ⎪ ⎨. ⎞⎫ ⎪ ⎪ n−1 n−1 ⎬ ⎟ ⎜ 2 2 ⎟ ⎜ λy (0) > h(y) 2(n − 1) H (y) − (y) κ (y) κ i j ⎠⎪ . ⎝ ⎪ ⎪ ⎪ i=1 j=1 ⎩ ⎭ ⎛. j

(93) =i. n = 2 S = γ(t), t ∈ [0, C] . λy (0) = h(γ(t))κ(t), ∀t ∈ [0, C]; λy (0) > 2h(γ(t))κ2 (t), ∀t ∈ [0, C]..

(94) Λ Λ (0) > 0. s=0. Λ (0) = 0. λy (s) . Λ(y , s) = λy (s). n−1

(95). (1 − sκi (y));. i=1. Λ (y , s) = λy (s). n−1

(96). ⎧ ⎪ n−1 ⎪ ⎨ . n−1

(97). ⎫ ⎪ ⎪ ⎬. (1 − sκj (y)) ; ⎪ ⎪ ⎪ ⎪ i=1 i=1 ⎩ j=1 ⎭ j

(98) =i ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ n−1 n−1 ⎨ n−1 ⎬

(99)

(100) −κ (y) Λ (y , s) = λy (s) (1 − sκi (y)) + 2λy (s) (1 − sκj (y)) ⎪ ⎪ i ⎪ i=1 i=1 ⎪ j=1 ⎭ ⎩ j

(101) =i ⎫ ⎧ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ n−1 n−1 n−1 ⎬ ⎬⎪ ⎨ ⎨

(102) + λy (s) −κj (y) . −κi (y) (1 − sκk (y)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ i=1 ⎩ j=1 k=1 ⎭ k

(103) =i,j (1 − sκi (y)) + λy (s). λy (0) = h(y). n−1 i=1. −κi (y). j

(104) =i. Λ (0) = 0. κi (y) = (n − 1)H(y). λy (0) = (n − 1)h(y)H(y). Λ (0) > 0,. ⎧ ⎪ ⎪ ⎨. ⎞⎫ ⎪ ⎪ n−1 n−1 ⎬ ⎟ ⎜ 2 2 ⎟ ⎜ λy (0) > h(y) 2(n − 1) H (y) − (y) κ (y) κ j ⎠⎪ . ⎝ i ⎪ ⎪ ⎪ i=1 j=1 ⎩ ⎭ ⎛. j

(105) =i. n = 2 S = γ(t), t ∈ [0, C] . λy (0) = h(γ(t))κ(t), ∀t ∈ [0, C]; λy (0) > 2h(γ(t))κ2 (t), ∀t ∈ [0, C]. (H1 ) n=2. n = 2, 3..

(106) S ⊂ Ω. S Sk ψ k : M k → Sk. S (k = 1, . . . , k0 ) (n − 1). Mk. Mk ⊂ Rn−1. Sk. M. J(ψk ) Mk ; k0 k=1. Sk = S M ⊂ R2. M. C1 n = 3. n=2. n=3 n = 2. M ⊂ R. n = 2. S S1. S S. S. S1 S = S1. S1 Sk (k = 1, . . . , k0 ) I k Sk J(ψk ) Ik . S2 = {(cos(s), sen(s)) : π < s < 2π}. Ik ⊂ R. ψk. S1 = {(cos(s), sen(s)) : 0 ≤ s ≤ π} ψk (s) = (cos(s), sen(s)) S = S1. n=2 n=3. S p≥0. p. p ≥ 0 p. S2. S p≥0. S. S. p S2.

(107) S 2. p. p=0. • S1 = {(x, y, z) ∈ S 2 : z ≥ ψ1−1 (x, y, z) = (x, y). √. 2 } 2. • S2 = {(x, y, z) ∈ S 2 : z ≤ − ψ2−1 (x, y, z) = (x, y). √. 2 } 2. M1 = {(x, y) ∈ R2 : x2 + y 2 ≤ 12 }. ψ1. M2 = {(x, y) ∈ R2 : x2 + y 2 ≤ 12 }. ψ2. √. √. • S3 = {(cos(s), sen(s), z) ∈ S 2 : 0 ≤ s ≤ π, − 22 < z < 22 } M3 = {(s, z) ∈ R2 : √ √ ψ3 (s, z) = (cos(s), sen(s), z) 0 ≤ s ≤ π, − 22 < z < 22 } √. √. • S4 = {(cos(s), sen(s), z) ∈ S 2 : π < s < 2π, − 22 < z < 22 } M4 = {(s, z) ∈ R2 : √ √ ψ4 (s, z) = (cos(s), sen(s), z) π < s < 2π, − 22 < z < 22 } p ≥ 1. p. 2p Sk , (k = 1, . . . , 4) Sk S1 × I S 3 S4. p=0. I. S. Ξ :S × Iδ → Sδ (y, s) → (y + sν(y)), Iδ = (−δ, δ) Sδ = {x ∈ Rn : dist(x, S) < δ}, ν(y) δ J(Ξ). y∈S. S δ>0. δ>0 Sk , (k = 1, . . . , k0 ) y ∈ Sk. S. s ∈ Iδ }. Ξ :Sk × Iδ → Sk,δ (y, s) → (y + sν(y)).. Sk,δ := {y + sν(y) :.

(108) ψk Mk , (k = 1, . . . , k0 ) ψ : Mk × I δ → Sk × I δ ψ(x , s) = (ψk (x ), s) J(ψ)(x , s) = J(ψk )(x ) (x , s) ∈ Mk × Iδ Ξk = Ξ ◦ ψ Ξk. . ψ J(Ξk ). Ξk : Mk × Iδ → Sk,δ (x , s) → (ψk (x ) + sν(ψ(x ))).. J(Ξk ) Mk := Ξ−1 k (Sk ) k0 . M. Sk = S. k=1. δ δ u 5 ∈ L (Sδ ) Ω = Sδ .. 5 u L1 (Sδ ). 1. Sδ ⊂ Ω u ∈ L1 (Ω) ≤ u L1 (Ω). u 5. Ω Sδ. ρ>0. 0 < v − v0 L1 (Sδ ) < ρ h|Dv| > Sδ k = (1, . . . , k0 ) v ∈ BV (Sk,δ ; {α, β}) 0 < h|Dv| > Sk,δ h|Dv0 | ρ = min{ρ1 , . . . , ρk0 } Sk,δ ρk. v ∈ BV (Sδ ; {α, β}. . h|Dv0 | ρk > 0. v − v0 L1 (Sk,δ ) < ρk Sδ. v0. . v0 (x , s) =. α, β,. s<0 s>0 (Sk,δ ) ∩ ∂∗ {v0 > ξ} = Sk. Sk,δ. h|Dv0 |. h|Dv0 | = Sk,δ. ∞ −∞ β. Sk,δ ∩∂∗ {v0 >ξ}. =. hdH α. Sk. n−1. hdHn−1 (x)dξ. (x)dξ = (β − α). hdHn−1 . Sk. ξ ∈ (α, β),.

(109) g˜˜ := g ◦ Ξ. M := Mk Sk,δ. g. g˜ := g ◦ Ξk ρk > 0. Jk,m Jk,m := inf{J(Ξk )(x , s) : (x , s) ∈ M × Iδ }.. a. ˜ J(Ξk ) h,. f. ˜ , 0)J(Ξk )(y , 0) 0 < r < ∞ supy ∈M h(y ˜ , s)J(Ξk )(y , s) r = 2r M < r inf (y ,s)∈M ×Iδ h(y ρ >0 ρk := min{Jk,m (β − α)ρ , 2δ Jk,m (β − α)Hn−1 (M )} v ∈ BV (Sk,δ ; {α, β} 0 < v − v0 L1 (Sk,δ ) < ρk h|Dv| > Sk,δ h|Dv0 | Sk,δ r. L1 (Sk,δ ). h|Dv| > (β − α) Sk,δ. s ∈ (−δ, δ). hdHn−1 . Sk. v˜(·, s). M s := M × {s}. • A+ v := {x ∈ M ×. %δ. & , δ : v˜(x) = α} 2. & % δ : v˜(x) = β}. := {x ∈ M × −δ, − • A− v 2 A− ρk v ⊂ {v

(110) = v0 }. Jk,m (β − α)ρ ≥ ρk > v − v0 L1 (Sk,δ ) = |v − v0 dLn Sk,δ. = |˜ v − v˜0 J(Ξk )dLn M ×Iδ ≥ |˜ v − v˜0 J(Ξk )dLn δ M ×( 2 ,δ ). |˜ v − v˜0 dLn ≥ Jk,m δ M ×( 2 ,δ ) δ n , δ : v(x , s) = α (x , s) ∈ M × = Jk,m (β − α)L 2 % & % +& n Av . = Jk,m (β − α)Ln A+ v = Jk,m (β − α)H A+ v ⊂ {v

(111) = v0 }. Hn (A+ v) < ρ A+ A−. Hn (A− v) < ρ.. A+ v. A− v.

(112) l∈. %δ. ,δ 2. &. & % l ∈ −δ, −δ 2. 1 n−1 Hn−1 (A+ (∂∗ (A+ v ∩ (M × {l})) ≤ H v ∩ (M × [l, δ]))\∂∗ (M × [l, δ])); r 1 n−1 Hn−1 (A+ (M ); v ∩ (M × {l})) ≤ H 4 1 n−1 Hn−1 (A− (∂∗ (A− v ∩ (M × {l })) ≤ H v ∩ (M × [−δ, l ])\∂∗ (M × [−δ, l ])); r 1 n−1 Hn−1 (A− (M ). v ∩ (M × {l })) ≤ H 4 l. l − Mv = {x ∈ M : (x , l) ∈ A+ v } ∪ {x ∈ M : (x , l ) ∈ Av },. Skv = Ξk (Mv ) A[l,δ] = ∂∗ (A+ v v ∩ (M × [l, δ]))\∂∗ (M × [l, δ]) . ] A[−δ,l = ∂∗ (A− v v ∩ (M × [−δ, l ])\∂∗ (M × [−δ, l ]). ( 1 ' n−1 [l,δ] n−1 [−δ,l ] (A ) + H (A ) H v v 2r 1 Hn−1 (Mv ) ≤ Hn−1 (M ). 2 Hn−1 (Mv ) ≤. (A). A E0. κi (y); (i = 1, . . . , n−1) S. (β − α). y∈S hdHn−1. (Ξ((Sk \Skv )×[l ,l]))∩∂∗ {v=α}. = (β − α). ((Sk \Skv )×[l ,l])∩∂∗ {v˜ ˜=α}. ˜˜ h. = (β − α). ((Sk \Skv )×[l ,l])∩∂∗ {v˜ ˜=α}. = (β − α). ≥ (β − α). ˜˜ n−1 hJ(Ξ)dH (. ((Sk \Skv )×[l ,l])∩∂∗ {v˜ ˜=α}. Sk \Skv. n−1

(113). (1 − sκi (y)) dHn−1 (. Λ(y, s)dHn−1 (. Λ(y, 0)dHn−1 (Λ. Sk \Skv. h(y)dHn−1 (. ). i=1. Λ) s = 0). = (β − α). Ξ). Λ)..

(114) (β − α). h(y)dHn−1. ˜ , 0)J(Ξk )(y , 0)dHn−1 ( = (β − α) h(y Skv. Ξk ). Mv. ˜ , 0)J(Ξk )(y , 0)Hn−1 (Mv ) ( ≤ (β − α) sup h(y. ). y ∈Mv. ˜ , 0)J(Ξk )(y , 0)Hn−1 (Mv ) (Mv ⊂ M ) ≤ (β − α) sup h(y y ∈M. ≤ (β − α)r ≤ (β − α)r. inf. (y ,s)∈M ×Iδ. ˜ , s)J(Ξk )(y , s)Hn−1 (Mv ) h(y. inf. (y ,s)∈M ×((−δ,l )∪(l,δ)). r ). ˜ , s)J(Ξk )(y , s)Hn−1 (Mv ) h(y. M × ((−δ, l ) ∪ (l, δ)) ⊂ M × Iδ ). (. # $ 1 ˜ , s)J(Ξk )(y , s) H(n−1) (A[l,δ] ) + H(n−1) (A[−δ,l ] ) inf h(y ≤ (β − α) v v (y ,s)∈M ×((−δ,l )∪(l,δ)) 2 1 ≤ (β − α) 2. ≤ (β − α). 5.13) [l,δ] [−δ,l ] Av ∪Av. [l,δ] [−δ,l ] Av ∪Av. ˜ , s)J(Ξk )(y , s)dHn−1 ( h(y. ˜ , s)J(Ξk )(y , s)dHn−1 h(y . (. Hn−1 (A[l,δ] ∪ Av[−δ,l ] ) > 0) v. = (β − α). ). (Ξk (M ×((−δ,l )∪(l,δ))))∩∂∗ {v=α}. hdHn−1 (. . Av[l,δ] ∪ Av[−δ,l ] ).. . Hn−1 (A[l,δ] ∪ Av[−δ,l ] ) > 0. v Λ(y, ·) y∈S s=0 v H ((Sk \Sk ) × [l , l]) ∩ ∂∗ {v˜˜ = α} = Sk \Skv v = α} = M \Mv [l , l]) ∩ ∂∗ {˜ n−1. . v˜(x , s) =. β, se (x , s) ∈ (M \Mv ) × (0, l) α, se (x , s) ∈ (M \Mv ) × (l , 0). ((M \Mv ) × Hn.

(115) v˜(x , s) =. α, se (x , s) ∈ (M \Mv ) × (0, l) β, se (x , s) ∈ (M \Mv ) × (l , 0).. ρk > v − v0 L1 (Sk,δ ) = |v − v0 dLn ( ρk · L1 (Sk,δ ) Sk,δ. |˜ v − v˜0 J(Ξk )dLn ( Ξk ) = M ×Iδ. |˜ v − v˜0 J(Ξk )dLn ( (M \Mv ) × (l , l) ⊂ M × Iδ ) ≥ (M \Mv )×(l ,l) |˜ v − v˜0 J(Ξk )dHn ( Ln H n ) =. ). (M \Mv )×(l ,l). ≥ Jk,m (β − α)Hn ((M \Mv ) × (l , l)) ( δ δ n ( ≥ Jk,m (β − α)H (M \Mv ) × − , 2 2 = Jk,m (β − α)δHn−1 (M \Mv ) δ ≥ Jk,m (β − α) Hn−1 (M ) ( 2 ≥ ρk (. . Jk,m ) δ δ − , ⊂ (l , l)) 2 2. ). ρk ).. ˜ v − v˜0 L1 (M \Mv ×[l ,l]) = 0. Hn−1 (Mv ) = 0 v ∈ BV (Sk,δ ; {α, β}) A+ v. |v − u0 |L1 ((−δ,δ)×M ) > 0. A− v − 0 < |˜ v − v˜0 |L1 (M ×((−δ,l ]∪[l,δ))) = (β − α)Hn (A+ v ∪ Av ).. − 1 Hn−2 (∂∗ ((A+ v ∪ Av ) ∩ (M × {t})\∂∗ (M × {t})) = 0 H. − 1 Hn−2 (∂∗ ((A+ v ∪ Av ) ∩ (M × {t})) = 0 H. − 1 Hn−1 (((A+ v ∪ Av ) ∩ ({t} × M )) = 0 H. (−δ, l ] ∪ [l, δ).. (−δ, l ] ∪ [l, δ). (−δ, l ] ∪ [l, δ).

(116) − Hn (A+ v ∪ Av ) = 0. Mv v˜(x , l) = v˜0 (x , l) = β. v˜(x , l ) = v˜0 (x , l ) = β. M \ Mv .. Skv v˜˜(x , l) = v˜˜0 (x , l) = β. v˜˜(x , l ) = v˜˜0 (x , l ) = β. Sk \ Skv .. Sk,δ Hn ⎧ ⎪ ξ ≤ α, ⎨ ∅ (Sk,δ ) ∩ ∂∗ {v > ξ} = (Sk,δ ) ∩ ∂∗ {v = α} α < ξ < β, ⎪ ⎩ β ≤ ξ. ∅ (Sk,δ ) ∩ ∂∗ {v = α}. (Sk,δ ) ∩ ∂∗ {v =. β}.. h|Dv| = Sk,δ. =. ∞. −∞ β. = α. h|Dv| (Skδ). (Sk,δ ∩∂∗ {v>ξ}. (Sk,δ )∩∂∗ {v=α}. ). hdHn−1 dξ (. = (β − α). hdHn−1 (Sk,δ )∩∂∗ {v=α}. = (β − α). hdHn−1 (Ξ((Sk. + (β − α). ≥ (β − α). (Ξk. )×[l ,l]))∩∂. ∗ {v=α}. (M ×((−δ,l )∪(l,δ))))∩{v=α}. hdHn−1 (. Ξ. hdHn−1 (Ξ((Sk \Skv )×[l ,l]))∩∂∗ {v=α}. + (β − α). hdHn−1 (Ξk. (M ×((−δ,l )∪(l,δ))))∩{v=α}. (Ξ((Sk \Skv ) × [l , l])) ⊂. (. > (β − α). (Ξ((Sk ) × [l , l]))). h(y)dHn−1. + (β − α). hdHn−1 dξ (. Sk \Skv. h(y)dHn−1 Skv. ). Ξk ).

(117) = (β − α). h(y)dHn−1 . Sk. u0 ⎧ ⎪ ⎨. E0. = 2 (a(x)∇u ) + f (x, u ), (t, x) ∈ R+ × Ω u (0, x) = φ(x), x ∈ Ω ⎪ ⎩ ∂u = 0, (t, x) ∈ R+ × ∂Ω. ∂ν ∂u ∂t. >0 Ω ν a α β f E : L1 (Ω) → R ∪ ∞, . a(x) |∇v|2 + 1 W (x, v) dx, Ω E (v) := ∞,. E0 S = ∂∗ {u0 = β} ∩ ∂∗ {u0 = α}. u0 u0 ∈ BV (Ω; {α, β}) S. v ∈ H 1 (Ω) E0. u0 S. S. u0. E0. u0 0 > 0. • v. {v }<0 L1 (Ω). • v − u0 L1 (Ω) → 0. E → 0. {v }. → 0. E. S. E. v v λ1 (v ) λ1 (v ) < 0. v v λ1 (v ) = 0 0. Ω W (v ) W (v ). 0 H (Ω) 1. v W (v ). v W (v ).

(118) v. E. v.

(119)

(120) L1 (Ω) δ > 0. M ⊂ R. Ω. Ω := M × Iδ M ⊂ R2 n=2. n−1. n=3. M ⊂R. E. Γ. E0. {v ∈ L (Ω) :. v dLn = m}. 1. Ω. m= Ω. Ln (Ω) (α 2 −. Ω. + β) Ω+. S. Ω− = {(x, s) ∈ M × (−δ, 0)}. Ω+ = {(x, s) ∈ M × (0, δ)}. ∂S ⊂ ∂Ω, ∂S ∩ ∂Ω = ∅.. S := M × {0}.

(121) E (v) := a. Ω. a(x) |∇v|2 + 1 F (x, v) dx,. v ∈ H 1 (Ω). ∞, g1 = α, g2 = β. F. E0 (v) =. Γ Ω. Ω. E0 h(x) |Dχv=β | ,. v ∈ BV (Ω; {α, β});. ∞. β. h(x) := 2. a(x)F (x, s)ds.. α. x ∈ S = M × {0} h(x , s). s=0 v0 (x) = αχΩ− (x) + βχΩ+ (x), x ∈ Ω E0. L1. ρk > 0 0 < v − v0 L1 (Sk,δ ) < ρk • M := Mk • ρ := ρk • S := Sk S = M × {0} • Ω = M × Iδ = S × Iδ := Sk,δ. L1 (Ω) Sk,δ := {y+sν(y) : y ∈ Sk s ∈ Iδ } v ∈ BV (Sk,δ ; {α, β}) h|Dv| > Sk,δ h|Dv0 | Sk,δ.

(122) • •. S. ν. S Λ. h S = M × {0} h . ρ. Ω. h|Dv| >. Λ v ∈ BV (Ω; {α, β}). Ω. 0 < v − v0 L1 (Ω) <. h|Dv0 | Ω. Ω =. M × Iδ Ω = Iδ × M Iδ x ∈ M f (x, u). h(·, ·). h. h(·, x ). M s = 0 f (x, u) := b(x)f (u) a b. h h(x) = K b(x). b > 0. C1 a. K. {v ∈ L (Ω) :. v dLn = m}. 1. Ω. m=. Ln (Ω) 2. (α + β) Γ. Ω. h(x , s). E0. v dLn = m x ∈ S = M × {0} s=0. v0 (x) = αχΩ− (x) + βχΩ+ (x), x ∈ Ω L1 E0 : {v ∈ L1 (Ω) : Ω v dLn = m} → R ∪ ∞ h(x) |Dχv=β | , v ∈ BV (Ω; {α, β}); Ω E0 (v) = ∞ β n h(x) := 2 α a(x)F (x, s)ds m = L 2(Ω) (α + β).

(123) m m= s1 =

(124) 0. E0 Ln (Ω) (α + 2. . v0 (x) = αχΩ− (x) + βχΩ+ (x) v dLn = m m Ω 0. β) s=0. v0. α, β, h(x , ·). s = s1 x ∈ S = M × {0} s=0. h(x , s) h. β. h(x) := 2. a(x)F (x, s)ds.. α. a. F E0. Ω. v dLn = m. . E0. ρ>0 0 < v − v0 L1 (Ω) < ρ. v ∈ {v ∈ L (Ω) : Ω v dL = m} h|Dv0 | E0 Ω 1 L (Ω) BV (Ω; {α, β} Ω ∩ ∂∗ {v0 > ξ} = M 1. n. h|Dv0 | = Ω. =. ∞. a. Ω. h|Dv| >. Ω. h|Dv0 |. −∞ Ω∩∂∗ {v0 >ξ} β n−1. hdH. α. v0 ξ ∈ (α, β),. . hdHn−1 (x)dξ. (x)dξ = (β − α). M. F. hdHn−1 . M. h r. 0 < r < ∞. supy ∈M h(y , 0) < r inf (y ,s)∈M ×Iδ h(y , s) M L1 (Ω) (A) A. r = 2r ρ > 0 ρ := (β − α)ρ.

(125) v ∈ BV (Ω; {α, β}) ∩ { Ω v dLn = m} 0 < v − v0 L1 (Ω) < ρ h|Dv| > Ω h|Dv0 | Ω. h|Dv| > (β − α) hdHn−1 . Ω. β}∩. (Ω). M. . ξ ∈ (α, β),. Ω ∩ ∂∗ {v > ξ} = ∂∗ {v = h|Dv| = (β−α) ∂∗ {v=β}∩int(Ω) hdHn−1 Ω . (β − α). hdH. n−1. ∂∗ {v=β}∩int(Ω). > (β − α). hdHn−1 . M. (β − α) > 0. hdH ∂∗ {v=β}∩int(Ω). s ∈ (−δ, δ) M = M × {s}.. n−1. hdHn−1 .. > M. v(·, s). s. def. • A+ v = {x ∈ M ×. %δ 2. & , δ : v(x) = α}. & % def δ • A− v = {x ∈ M × −δ, − 2 : v(x) = β}. A+ v ⊂ {v

(126) = v0 }. ρ. (β − α)ρ ≥ ρ > v − v0 L1 (Ω) = |v − v0 |dLn Ω. = |v − v0 |)dLn M ×Iδ ≥ |v − v0 dLn δ M ×( 2 ,δ ) δ n , δ : v(x , s) = α (x , s) ∈ M × = (β − α)L 2 % & % +& n Av . = (β − α)Ln A+ v = (β − α)H Hn (A+ v) < ρ Hn (A− v) < ρ.. A+ v. A− v ⊂ {v

(127) = v0 } − Av A+ A− l∈. %δ. ,δ 2. &. & % l ∈ −δ, −δ 2. 1 n−1 Hn−1 (A+ (∂∗ (A+ v ∩ (M × {l})) ≤ H v ∩ (M × [l, δ]))\∂∗ (M × [l, δ])); r 1 n−1 Hn−1 (A− (∂∗ (A− v ∩ (M × {l })) ≤ H v ∩ (M × [−δ, l ])\∂∗ (M × [−δ, l ])). r.

(128) l. l − Mv = {x ∈ M : (x , l) ∈ A+ v } ∪ {x ∈ M : (x , l ) ∈ Av },. A[l,δ] = ∂∗ (A+ v v ∩ (M × [l, δ]))\∂∗ (M × [l, δ]) . ] A[−δ,l = ∂∗ (A− v v ∩ (M × [−δ, l ])\∂∗ (M × [−δ, l ]). Hn−1 (Mv ) ≤. E0. ( 1 ' n−1 [l,δ] n−1 [−δ,l ] (A ) + H (A ) . H v v 2r. hdHn−1 ((M \Mv ≥. =. M \Mv. M \Mv. )×[l ,l])∩∂. ∗ {v=α}. h(y , 0)dHn−1 (h. s = 0) y := (y , 0)).. h(y)dHn−1 (. h(y)dHn−1. Mv. h(y , 0))dHn−1 (. =. (y , 0) := y). Mv. ≤ sup h(y , 0))Hn−1 (Mv ) ( y ∈Mv. ). ≤ sup h(y , 0))Hn−1 (Mv ) (Mv ⊂ M ) y ∈M. ≤ r ≤ r. inf. (y ,s)∈M ×Iδ. inf. (y ,s)∈M ×((−δ,l )∪(l,δ)). ( ≤. h(y , s)Hn−1 (Mv ). r ). h(y , s)Hn−1 (Mv ). M × ((−δ, l ) ∪ (l, δ)) ⊂ M × Iδ ) # $ inf h(y , s) H(n−1) (Av[l,δ] ) + H(n−1) (Av[−δ,l ] ). 1 2 (y ,s)∈M ×((−δ,l )∪(l,δ)). 6.6). 1 h(y , s)dHn−1 ( [−δ,l ] 2 A[l,δ] v ∪Av 1 = hdHn−1 ( 2 ((M ×((−δ,l )∪(l,δ))))∩∂∗ {v=α}. hdHn−1 . ≤ ≤. ((M ×((−δ,l )∪(l,δ))))∩∂∗ {v=α}. ) . A[l,δ] ∪ Av[−δ,l ] ) v.

(129) ∂∗ {v = β} ∩ (. (M \Mv ) × (l , l)). M \Mv. Hn−1 (Mv ) > 0; Hn−1 (Mv ) = 0. ∂∗ {v = β}∩( (M \Mv )×(l , l)) M \Mv . v − v0 L1 (Ω) > 0 v dx = m Ω n H ({v

(130) = v0 } ∩ (M × ((−δ, l ) ∪ (l, δ)))) > 0 Mv Hn−1 (∂∗ {v = β} ∩ (M × ((−δ, l ) ∪ (l, δ)))) > 0 v dx = m Ω. hdHn−1 ∂∗ {v=β}∩int(Ω). =. hdHn−1 ∂∗ {v=β}∩int(M ×((l ,l)∪((−δ,l )∪(l,δ))). ≥. hdHn−1 ∂∗ {v=β}∩. +. ((M \Mv )×(l ,l)). hdHn−1 ∂. ∗. {v=β}∩int(M ×((−δ,l )∪(l,δ))). >. hdH. n−1. M \Mv. =. +. hdHn−1 Mv. hdHn−1 , M. h(y)dHn−1 Mv. 1 ≤ hdHn−1 2 ((M ×((−δ,l )∪(l,δ))))∩{v=α}. < hdHn−1 . ((M ×((−δ,l )∪(l,δ))))∩{v=α}. hdHn−1 ∂∗ {v=β}∩int(Ω). =. hdHn−1 ∂∗. {v=β}∩int(M ×((l ,l)∪((−δ,l )∪(l,δ))).

(131) ≥. hdHn−1. ∂∗ {v=β}∩. +. ((M \Mv )×(l ,l)). hdHn−1 ∂. ∗. {v=β}∩int(M ×((−δ,l )∪(l,δ))). hdHn−1. > )×(l ,l)). ∂∗ {v=β}∩ ((M \Mv. 1 + hdHn−1 2 ∂∗ {v=β}∩int(M ×((−δ,l )∪(l,δ))). n−1 ≥ hdH + hdHn−1 M \Mv Mv. = hdHn−1 . M. Hn−1 (Mv ) > 0 0. Hn−1 (∂∗ {vβ}∩(. hdH ∂∗ {v=β}∩. n−1. =. (Ω). +. hdHn−1 ∂∗ {v=β}∩(. (M )×((−δ,l )∪(l,δ))). hdHn−1 ∂∗ {v=β}∩(. (M )×(l ,l)). hdHn−1. > (M )×(l ,l)). ∂∗ {v=β}∩( = hdHn−1 , M. Hn−1 (Mv ) = 0. (M )×((−l, l )∪(l, l))) >.

(132)

(133)

(134)