Entropy Solutions of Doubly Nonlinear Fractional Laplace Equations

In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $$\mathbb {R}^N$$ R N . The equation is driven by the fractional Laplacian $$(-\varDelta )^{\frac{s}{2}}$$ ( - Δ ) s 2 for $$s\in (0,1]$$ s ∈ ( 0 , 1 ] and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $$L^1$$ L 1 -contraction and comparison principle and to show existence and uniqueness of entropy solutions.

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ordinary differential equation (ODE) for the different phases of steel, and Maxwell’s equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e. that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g. it allows to include free energy functions with low regularity properties corresponding to phase transitions.