Related papers: Non-local self-improving properties: A functional …
We consider the problem of estimating the value of a linear functional in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The…
We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with…
This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic…
We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is…
We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be…
In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic…
In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates…
This paper completes and partially improves some of the results of [arXiv:0809.5002] about the asymptotic behavior of solutions of linear and nonlinear elliptic equations with singular coefficients via an Almgren type monotonicity formula
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent…
Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated…
We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element…
We study partial analyticity of solutions to elliptic systems and analyticity of level sets of solutions to nonlinear elliptic systems. We consider several applications, including analyticity of flow lines for bounded stationary solutions…
This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as…
In this work we establish local $C^{2,\alpha}$ regularity estimates for flat solutions to non-convex fully nonlinear elliptic equations provided the coefficients and the source function are of class $C^{0,\alpha}$. For problems with merely…
This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform…
Fractional diffusion has become a fundamental tool for the modeling of multiscale and heterogeneous phenomena. However, due to its nonlocal nature, its accurate numerical approximation is delicate. We survey our research program on the…
This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass…
We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with time-independent coefficients and apply them to obtain observability inequalities for its solutions over measurable sets.