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On a class of weighted anisotropic Sobolev inequalities
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TL;DR: In this paper, a weighted anisotropic Sobolev type inequality was proposed for finite cylinders, where the weights being different powers of the distance function from the top and the bottom of the cylinder.
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Abstract: In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities, that is Sobolev type inequalities where different derivatives have different weight functions. The inequalities we are dealing with, are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider here Sobolev inequalities on finite cylinders, the weights being different powers of the distance function from the top and the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is the distance function from an higher codimension part of the boundary.
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Citations
Improving L2 estimates to Harnack inequalities
TL;DR: In this article, the authors consider operators of the form L = −L − V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain with Dirichlet boundary conditions.
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Symmetry breaking in semilinear boundary value problems
miliredon1981
- 29 Sep 2022
TL;DR: In this article , the authors studied symmetry breaking for semilinear partial differential equations, which encompasses equations whose symmetries are not necessarily inherited by their solutions, which is particularly interesting for ground state solutions.
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References
The Liouville Property and a Conjecture of De Giorgi
TL;DR: In this article, the authors consider bounded entire solutions of the nonlinear PDE Δu + u u 3 = 0i n R d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes.
166
On the Grushin operator and hyperbolic symmetry
William Beckner
- 10 Oct 2000
TL;DR: In this article, the complexity of geometric symmetry for differential operators with mixed homogeneity is examined and sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.
•Book
Weighted Sobolev Spaces
Alois Kufner
- 23 Jul 1985
TL;DR: In this article, the power-type weights and general weights were investigated, and several elementary results were given for different power types and the density of Smooth Functions Imbedding Theorems.
Phase transitions: Uniform regularity of the intermediate layers
Luis A. Caffarelli,Antonio Có +1 more
TL;DR: In this paper, the authors considered the problem of energy minimization of convex functionals in the calculus of variations, where the nonnegative function F is a double well potential vanishing only for two values of u, and the minimizers u under consideration will take values precissely at that interval (−1 ≤ u ≤ + 1).
Sobolev Inequalities for Weighted Gradients
TL;DR: In this paper, the symmetry, existence, and uniqueness properties of extremal functions for the weighted Sobolev inequality were studied for the case where x ∈ m, y ∈ k with m,k ≤ 1, α ≤ 0, and Q ≥ 1.