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Cited by 2 publications
References 17 publications
Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains
For bounded domains $$\Omega $$ Ω with Lipschitz boundary $$\Gamma $$ Γ , we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type $$\begin{aligned} {\mathcal {A}}=\begin{pmatrix} B^*B & 0 \\ -{\mathscr {N}}^{\mathfrak {b}}B & \gamma \end{pmatrix}, \end{aligned}$$ A = B ∗ B 0 - N b B γ , where B is associated to a quadratic form $${\mathfrak {b}}$$ b and $${\mathscr {N}}^{{\mathfrak {b}}}$$ N b an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space $${\mathcal {H}}:=L^2(\Omega ) \times L^2(\Gamma )$$ H : = L 2 ( Ω ) × L 2 ( Γ ) . Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form $$B=-\mathop {{div} }Q \nabla $$ B = - div Q ∇ with $$Q \in C^{1,1}({\overline{\Omega }},{\mathbb {R}}^{d\times d}),$$ Q ∈ C 1 , 1 ( Ω ¯ , R d × d ) , also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.
Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains
For bounded domains $$\Omega $$ Ω with Lipschitz boundary $$\Gamma $$ Γ , we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type $$\begin{aligned} {\mathcal {A}}=\begin{pmatrix} B^*B & 0 \\ -{\mathscr {N}}^{\mathfrak {b}}B & \gamma \end{pmatrix}, \end{aligned}$$ A = B ∗ B 0 - N b B γ , where B is associated to a quadratic form $${\mathfrak {b}}$$ b and $${\mathscr {N}}^{{\mathfrak {b}}}$$ N b an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space $${\mathcal {H}}:=L^2(\Omega ) \times L^2(\Gamma )$$ H : = L 2 ( Ω ) × L 2 ( Γ ) . Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form $$B=-\mathop {{div} }Q \nabla $$ B = - div Q ∇ with $$Q \in C^{1,1}({\overline{\Omega }},{\mathbb {R}}^{d\times d}),$$ Q ∈ C 1 , 1 ( Ω ¯ , R d × d ) , also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.
Еліптичні Оператори І Крайові Задачі У Просторах Узагальненої Гладкості
УДК 517.518.2+517.956.22 Наведено огляд результатів, отриманих протягом останніх десяти років у розробленій авторами теорії еліптичних крайових задач у функціональних просторах Хермандера, та пов'язані з ними інші результати сучасного аналізу. Основи цієї теорії та деякі її застосування систематично викладено у монографії „Hörmander spaces, interpolation, and elliptic problems'' (De Gruyter, Berlin/Boston, 2014) перших двох авторів огляду.
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