Search results for "p-Laplace equation"
showing 6 items of 6 documents
Inverse problems for $p$-Laplace type equations under monotonicity assumptions
2016
We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $\sigma_1 \geq \sigma_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.
Monotonicity and enclosure methods for the p-Laplace equation
2018
We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.
Enclosure method for the p-Laplace equation
2014
We study the enclosure method for the p-Calder\'on problem, which is a nonlinear generalization of the inverse conductivity problem due to Calder\'on that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.
Quantitative uniqueness estimates for pp-Laplace type equations in the plane
2016
Abstract In this article our main concern is to prove the quantitative unique estimates for the p -Laplace equation, 1 p ∞ , with a locally Lipschitz drift in the plane. To be more precise, let u ∈ W l o c 1 , p ( R 2 ) be a nontrivial weak solution to div ( | ∇ u | p − 2 ∇ u ) + W ⋅ ( | ∇ u | p − 2 ∇ u ) = 0 in R 2 , where W is a locally Lipschitz real vector satisfying ‖ W ‖ L q ( R 2 ) ≤ M for q ≥ max { p , 2 } . Assume that u satisfies certain a priori assumption at 0. For q > max { p , 2 } or q = p > 2 , if ‖ u ‖ L ∞ ( R 2 ) ≤ C 0 , then u satisfies the following asymptotic estimates at R ≫ 1 inf | z 0 | = R sup | z − z 0 | 1 | u ( z ) | ≥ e − C R 1 − 2 q log R , where C > 0 depends …
Calderón's problem for p-laplace type equations
2016
We investigate a generalization of Calderón’s problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation div σ |∇u|p−2 ∇u = 0 with 1 < p < ∞, which reduces to the standard conductivity equation when p = 2. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity σ may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth conductivity on the boundary. We use the enclosure method of Ikehata to recover the…
On the Porosity of Free Boundaries in Degenerate Variational Inequalities
2000
Abstract In this note we consider a certain degenerate variational problem with constraint identically zero. The exact growth of the solution near the free boundary is established. A consequence of this is that the free boundary is porous and therefore its Hausdorff dimension is less than N and hence it is of Lebesgue measure zero.