اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn epiperimetric inequality approach to the parabolic Signorini problem
104
0
0.0
(
0
)
تأليف
Wenhui Shi
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the vanishing order of a solution at a free boundary point is close to $3/2$ or an even integer, then the solution is asymptotically homogeneous. Furthermore, one can derive a convergence rate estimate towards the asymptotic homogeneous solution. As a consequence, we obtain the regularity of the regular free boundary as well as the frequency gap.
قيم البحث
اقرأ أيضاً
We investigate the regularity of the free boundary for the Signorini problem in $mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^infty$. However, even for $C^infty$ obstacles $varphi$, the set of non-regular (or degen
erate) points could be very large, e.g. with infinite $mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when $varphi$ is analytic, and when $Deltavarphi < 0$. However, even in these cases, the set of degenerate points is in general $(n-1)$-dimensional (as large as the set of regular points). In this work, we show for the first time that, usually, the set of degenerate points is small. Namely, we prove that, given any $C^infty$ obstacle, for almost every solution the non-regular part of the free boundary is at most $(n-2)$-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $(-Delta)^s$, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $(n-1-alpha_circ)$-dimensional for almost all times $t$, for some $alpha_circ > 0$. Finally, we construct some new examples of free boundaries with degenerate points.
In this note we discuss the (higher) regularity properties of the Signorini problem for the homogeneous, isotropic Lame system. Relying on an observation by Schumann cite{Schumann1}, we reduce the question of the solutions and the free boundary regul
arity for the homogeneous, isotropic Lame system to the corresponding regularity properties of the obstacle problem for the half-Laplacian.
We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace--Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method
based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces.
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional c
urvature of the underlying manifold is nonnegative. Later, Kim improved Cabres result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.
In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincare inequality. We prove a location and scale invariant Harnack inequality f
or a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigoryan--Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
