Use cases of Degenerate Elliptic Equations part5(Machine Learning 2024)
- Matrix Weights and Regularity for Degenerate Elliptic Equations(arXiv)
Author : Giuseppe Di Fazio, Maria Stella Fanciullo, Dario Daniele Monticelli, Scott Rodney, Pietro Zamboni
Abstract : We prove local boundedness, Harnack’s inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev-Poincaré inequalities. Data integrability is close to L1 and is exploited in terms of a suitable Stummel-Kato class that in some cases is necessary for local regularity.
2. Convergence at infinity for solutions of nonhomogeneous degenerate elliptic equations in exterior domains(arXiv)
Author : Leonardo Prange Bonorino, Lucas Pinto Dutra, Filipe Jung dos Santos
Abstract : In this work, first we prove that for any compact set K⊂Rn and any continuous function φ defined on ∂K, there exists a bounded weak solution in C(Rn∖K¯)∩C1(Rn∖K) to the exterior Dirichlet problem
{−div(|∇u|p−2A(|∇u|)∇u)=f in Rn∖Ku=φ on ∂K
provided p>n, A satisfies some growth conditions and f∈L∞(Rn) meets a suitable pointwise decay rate. We obtain thereafter the existence of the limit at the infinity for solutions to this problem, for any p∈(1,+∞) and n≥2. Moreover, for p>n we can show that the solutions converge at some rate and for p<n the convergence holds even for some unbounded f.
