Related papers: The Dirichlet problem for the Bellman equation at …

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $\Omega$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such…

In this paper we show how a second order scalar uniformly elliptic equation on divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary…

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The…

Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose…

We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural…

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions,…

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second…

A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ \hbox{ in }\Omega,…

Consider the eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \varphi -2\alpha\nabla m(x)\cdot \nabla\varphi+V(x)\varphi=\lambda\varphi\ \ \hbox{ in }\Omega, \end{equation} complemented by…

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural…

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a…

In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,\beta}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using…

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin.…

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral…

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where $\rho$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in…

We establish that the Dirichlet problem for convex linear growth functionals on $BD$, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $C^{1,\alpha}$-regularity theory as presently available for…

We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be…

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…