Abstract
We discuss the H¨older continuity property for the inverse mapping that identifies the diffusivity matrix A(x) in the main part of anisotropic p-Laplace equation as a function of resolvent operator. In particular, we prove that, within a chosen class of non-smooth admissible matrices the resolvent determines the anisotropic diffusivity in a unique manner and the correspondent inverse mapping is H¨older continuous in suitable topologies.
Highlights
Throughout the paper Ω is a bounded open subset of RN, N 2, for whichPoincare’s inequality holds, p1/p + 1/q = 1
We prove that the inverse mapping from resolvent to the matrix A is Holder continuous in suitable topologies
For p = 2, for p ∈ (2; 4], for p > 4 where the weighted Sobolev space HAp (Ω) is defined in (3.1), by RA we denote the inverse or resolvent operator for problem (1.3)–(1.4), which is uniquely determined by the matrix A ∈ M(Ω) and maps continuously HAp (Ω) ∗ into HAp (Ω), while the p value 1 −
Summary
Our main goal is to analyze the inverse problem of identifying the matrix of anisotropic diffusivity A(x) in the principle part of quasi-linear elliptic equation (1.3) as a function of resolvent operator. We prove that, within the class of admissible matrices M(Ω), the resolvent determines the anisotropic diffusivity in a unique manner. For p = 2, for p ∈ (2; 4], for p > 4 where the weighted Sobolev space HAp (Ω) is defined in (3.1), by RA we denote the inverse or resolvent operator for problem (1.3)–(1.4), which is uniquely determined by the matrix A ∈ M(Ω) and maps continuously HAp (Ω) ∗ into HAp (Ω), while the p value 1 −. It is worth to notice that, by analogy with a linear case [9], this result plays a key role when applying greedy algorithms to the approximation of parameter-dependent quasi-linear elliptic problems with anisotropic p-laplacian in an uniform and robust manner, independent of the given source terms (see, for instance, [2,3,4])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
