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Register a new userAnalysis and boundary value problems on singular domains: an approach via bounded geometry
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We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.
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We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $mathbb{C}$ that depend smoothly on a real parameter $lambda in [0,1]$. We derive the precise regularity of the solutions in all variables, including the parameter $lambda$. More specifically we show that the solution and its derivatives are continuous in all variables, and the Holder norms of the space variables are bounded uniformly in $lambda$.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with $L^2$ boundary data in a periodically perforated Lipschitz domain.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $F subset partial Omega$ be a $C^2$ submanifold of dimension $0 leq k leq N-2$. Put $delta_F(x)=dist(x,F)$, $V=delta_F^{-2}$ in $Omega$ and $L_{gamma V}=Delta + gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $Omega$. We study positive solutions of equations (LE) $-L_{gamma V} u = 0$ and (NE) $-L_{gamma V} u+ f(u) = 0$ in $Omega$ when $gamma < C_H(V)$ and $f in C({mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=partial Omega$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $Omega$ and establish existence and stability results when the data is concentrated on the set of subcritical points.
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W^1_p$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the $W^1_p$ and weighted $W^1_p$ estimates for the Robin problem in these domains.
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