相关论文: On the logarithmic epiperimetric inequality for th…
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
We consider the obstacle problem with irregular barriers for semilinear elliptic equation involving measure data and operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well…
In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of…
Let $({\M}, g(t))$ be a K\"ahler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on…
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our…
Using the epiperimetric inequalities approach, we study the obstacle problem $\min\{(-\Delta)^su,u-\varphi\}=0,$ for the fractional Laplacian $(-\Delta)^s$ with obstacle $\varphi\in C^{k,\gamma}(\mathbb{R}^n)$, $k\ge2$ and $\gamma\in(0,1)$.…
We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a…
The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same…
We study the inverse problem of determining a Signorini obstacle from boundary measurements for the isotropic elasticity system. We prove that the obstacle can be uniquely determined by a single measurement of displacement and normal stress…
We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale…
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.
An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness…
The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the…
We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.
We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the…
We establish symmetry results for two categories of overdetermined obstacle problems: a Serrin-type problem and a two-phase problem under the overdetermination that the interface serves as a level surface of the solution. The first proof…
We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational…