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In this work, we study the existence and nonexistence of solution for strongly coupled elliptic systems to m-parameters.
We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet…
This paper is devoted to establishing results for semilinear elliptic boundary value problems where the solvability of problems subject to {\it No Flux} boundary conditions follows from the solvability of related {\it Dirichlet} boundary…
We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
We develop a boundary integral equation solver for elliptic partial differential equations on complex \threed geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular…
In the thesis at hand we give a comprehensive discussion of basic problems for generalized Maxwell equations with mixed boundary conditions using the calculus of alternating differential forms on Riemannian manifolds of arbitrary dimension.…
The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^\lambda v$ is reduced to…
We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary…
We consider an elliptic operator $L$ with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to $Lu=0$ that attain given Neumann and Dirichlet-regularity data on different parts of the boundary.…
The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The numerical implementation of this method to linear elliptic PDEs formulated in the {\it interior} of a…
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary H\"{o}lder regularity under proper…
We give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent.
Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we…
We consider an elliptic pseudo differential equation in a multi-dimensional cone and starting wave factorization concept we add some boundary conditions. For the simplest cases explicit formulas for solution are given like layer potentials…
We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over…
We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of…
Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve…