Related papers: A logarithmic epiperimetric inequality for the obs…
Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with…
We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after…
A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain…
This preprint is a text for students and teachers on inequalities. Some standard topics are covered on application of calculus to inequality proving. Many examples are considered, stated, solved or partially solved. Some problems are…
We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…
We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash…
The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function…
The subject of these Notes is the new proof, proposed in [F. H{\'e}lein, In{\'e}galit{\'e} isop{\'e}rim{\'e}trique et calibrations, Annales de l'Institut Fourier 44, 4 (1994), 1211-1218] of the classical isoperimetric inequality in the…
The article studies the reiterated homogenization of linear elliptic variational inequalities arising in problems with unilateral constrains. We assume that the coefficients of the equations satisfy and abstract hypothesis covering on each…
In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the…
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…
A method for proving symmetrization inequalities for some elliptic p.d.e.'s on manifolds equipped with appropriate isoperimetric inequalities is outlined. The method is based on a modification of an approach of Baernstein. The question of…
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…
We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…
We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable…
We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that…
We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality…
In this paper, a class of smoothing modulus-based iterative method was presented for solving implicit complementarity problems. The main idea was to transform the implicit complementarity problem into an equivalent implicit fixed-point…