6533b7d7fe1ef96bd1268dd7

RESEARCH PRODUCT

Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions

José M. MazónN. IgbidaFuensanta AndreuJulián Toledo

subject

PhysicsElliptic operatorNonlinear systemPure mathematicsElliptic partial differential equationBounded functionStefan problemBoundary value problemUniquenessWeak formulation

description

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du)+γ(u) ∋ φ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x, Du)·η+β(u) ∋ ψ. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Δp(u) = div (|Du|p−2Du), with p > 1, γ and β are maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) and 0 ∈ β(0), and the data φ ∈ L1 (Ω) and ψ ∈ L1 (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem.

https://doi.org/10.1007/978-3-7643-7719-9_2