The mixed problem for harmonic functions in polyhedra

Mixed boundary value problem for $p$-harmonic functions in an infinite cylinder

We study a mixed boundary value problem for the $p$-Laplace equation $Δ_p u=0$ in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. Existence of weak solutions to the mixed problem is proved both for Sobolev and for continuous data on the Dirichlet part of the boundary. We also obtain a boundary regularity result for the point at infinity in terms of a variational capacity adapted to the cylinder.

Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^α_{τ,τ}, \ \frac{1}τ=\fracα{d}+\frac{1}{p}\ $ of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractiona…

关于混合边值问题的$W^{2,p}$可解性

On the $W^{2,p}$ solvability for mixed boundary value problems

本文建立了混合Dirichlet-Neumann边界条件下Poisson方程在两类区域中的全局$W^{2,p}$可解性，获得了最优参数范围，并推广到Lamé系统。

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.

多面体域中Ventcel问题的正则性及先验误差分析

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

本文研究了多面体域中Laplace算子混合边界值问题的正则性及有限元近似误差分析，通过分解解为正则和奇异部分，利用插值估计在域内和Ventcel面上进行先验误差分析。

具有混合边界条件的薛定谔算子特征值不等式

Inequalities for eigenvalues of Schrödinger operators with mixed boundary conditions

本文研究有界凸区域上带混合边界条件的薛定谔算子特征值问题，证明了不同边界条件选择下最低特征值之间的不等式，以及多维多面体区域上高阶混合特征值与纯狄利克雷特征值之间的不等式。

Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replacements for Dirichlet-Neumann fields. Moreover, we even show convergence in operator norm.

Time-Harmonic Electro-Magnetic Scattering in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell's equations in an exterior weak Lipschitz domain divided into two disjoint weak Lipschitz parts We will present a solution theory using the framework of polynomially weighted Sobolev spaces for the rotation and divergence. We will show a Fredholm alternative type result to hold using the principle of limiting absorption introduced by Eidus in the 1960's. The necessary a-priori-estimate and polynomial decay of eigenfunctions for the Maxwell equations will be obtained by transferring well kno…