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We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles…
In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \…
In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u =…
In this paper, we study the homogenization of elliptic equations that combine a local part, given by the Laplacian with Neumann boundary conditions, and its nonlocal version, defined through an integral operator with a smooth kernel. These…
In this note we devise and analyze a well-posed variational formulation of the Neumann boundary value problem associated to the biharmonic operator $\Delta^2$. An alternative formulation as a system of two Poisson problems for the Laplace…
This work revolves around the rigorous asymptotic analysis of models in nonlocal hyperelasticity. The corresponding variational problems involve integral functionals depending on nonlocal gradients with a finite interaction range $\delta$,…
The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the L\'{e}vy processes and stochastic interfaces. This paper deals with the nonlocal…
We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear…
On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We…
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional…
In this work, we consider a boundary value problem for nonlinear triharmonic equation. Due to the reduction of nonlinear boundary value problems to operator equation for nonlinear terms we establish the existence, uniqueness and positivity…
By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical…
We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…
In this paper we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic…
We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence,…
A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calder\'on-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish…
Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also…
The transparent boundary condition for the free Schr\"{o}dinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_{\Gamma}}$ where $\triangle_{\Gamma}$ is the…
We apply a recently proposed approximation method to the evaluation of non-Gaussian integral and anharmonic oscillator. The method makes use of the truncated perturbation series by recasting it via the modified Laplace integral…
We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter,…