Related papers: On the logarithmic epiperimetric inequality for th…
For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23--50). In particular at singular points we introduce a…
In this paper we give a proof of an epiperimetric inequality in the setting of the lower dimensional obstacle problem. The inequality was introduced by Weiss (Invent. Math., 138 (1999), no. 1, 23-50) for the classical obstacle problem and…
We prove an epiperimetric inequality for the thin obstacle problem, extending the pioneering results by Weiss on the classical obstacle problem (Invent. Math. 138 (1999), no. 1, 23-50). This inequality provides the means to study the rate…
We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: as the radius increases energy decays while negative and…
In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior…
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous…
We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric…
We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Lojasiewicz inequality. The difficulty lies in…
The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability…
Existence of solutions to an obstacle $p$-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally…
We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering…
We study the obstacle problem for the Evolutionary p-Laplace Equation when the obstacle is discontinuous and without regularity in the time variable. Two quite different procedures yield the same solution.
In this note, we study an obstacle problem for the elastic flow. We prove the local-in-time existence of weak solutions and discuss their relation to classical solutions when additional regularity is obtained. Related results concerning…
We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of…
In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…
We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In…
We discuss some regularity issues in the study of the obstacle problem. In particular, we present a recent result by O. Savin and the author on the regularity of the singular set for the obstacle problem with a fully nonlinear elliptic…
For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type.…
We study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and…
The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data.…