Maximal L2-regularity in nonlinear gradient systems and perturbations of sublinear growth

Abstract

Introduction 1 2. Preliminaries 2 3. Main result 5 4. Proof of the main result 8 5. Application to j-elliptic functions 10 Appendix A. Brézis' maximal L 2 -regularity theorem 13 References 17ABSTRACT. The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function ϕ has a smoothing effect, discovered by H. Brézis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of ϕ are not only closed, but even compact. We apply our results to the p-Laplacian and also to the Dirichlet-to-Neumann operator with respect to p-harmonic functions.

A doubly nonlinear evolution problem involving the fractional p-Laplacian

“…For this class of operators A = ∂E, the Cauchy problem (2.2) has the smoothing effect that every mild solution u of (2.2) is strong. This result is due to Brezis [Bre71] (see also [AH20]).…”

“…, and see also [CHK16,AH20]). For this class of operators A = ∂E, the Cauchy problem (2.2) has the smoothing effect that every mild solution u of (2.2) is strong.…”

A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation

“…A detailed verification of the claims of the example can be found in "Appendix 6". Essentially, (1) shows that in the unperturbed setting the origin is a singular state, which is due to the singularity of b in zero. In contrast to this, the presence of the highly oscillating path w H (ω) ensures that the solution u does not spend too much time in the singularity of b.…”

“…The famous theory of monotone operators traces back to the early works of Minty [32] and Browder [5]. It inspired many mathematicians to study well-posedness of monotone evolution equations and perturbations of it, see, e.g., [1,8,28,29,31,34,35]. However, if the potential b(u) cannot be treated as a compact perturbation, wellposedness breaks down and solutions may blow up in finite time, see, e.g., [12,38].…”

“…From a physical point of view, translating the potential in time can be interpreted as uncertainty as to the location of the origin of the potential. By a "regularizing path", we will understand a continuous path w that admits a sufficiently regular local time 1 such that techniques from pathwise regularization by noise à la [7,16,27] become applicable. Let us briefly sketch some main ideas of these approaches, in particular, in the spirit of [27], which we will be able to employ also in the study of (1.2).…”