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Register a new userPoincare Inequalities and Neumann Problems for the Variable Exponent Setting
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We extend the results of [5], where we proved an equivalence between weighted Poincare inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincare inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
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We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
We consider well-posedness of the boundary value problem in presence of an inclusion with complex conductivity $k$. We first consider the transmission problem in $mathbb{R}^d$ and characterize solvability of the problem in terms of the spectrum of the Neumann-Poincare operator. We then deal with the boundary value problem and show that the solution is bounded in its $H^1$-norm uniformly in $k$ as long as $k$ is at some distance from a closed interval in the negative real axis. We then show with an estimate that the solution depends on $k$ in its $H^1$-norm Lipschitz continuously. We finally show that the boundary perturbation formula in presence of a diametrically small inclusion is valid uniformly in $k$ away from the closed interval mentioned before. The results for the single inclusion case are extended to the case when there are multiple inclusions with different complex conductivities: We first obtain a complete characterization of solvability when inclusions consist of two disjoint disks and then prove solvability and uniform estimates when imaginary parts of conductivities have the same signs. The results are obtained using the spectral property of the associated Neumann-Poincare operator and the spectral resolution.
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
The Neumann-Poincare operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincare operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincare operator for curves of class $C^{2,alpha}$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a $C^{2,alpha}$ curve is perturbed by inserting a small corner.
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: $$-triangle u +operatorname{div}(umathbf{b}) =f quadtext{ and }quad -triangle v -mathbf{b} cdot abla v =g$$ in a bounded Lipschitz domain $Omega$ in $mathbb{R}^n$ $(ngeq 3)$, where $mathbf{b}:Omega rightarrow mathbb{R}^n$ is a given vector field. Under the assumption that $mathbf{b} in L^{n}(Omega)^n$, we first establish existence and uniqueness of solutions in $L_{alpha}^{p}(Omega)$ for the Dirichlet and Neumann problems. Here $L_{alpha}^{p}(Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(partialOmega)$.
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