The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems

Hauer, Daniel; Mazón, José M.

doi:10.1007/s00526-021-02149-5

Cited by 7 publications

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“…In the previous examples (cf Examples 2.5 and Example 3.6), V is a Banach space injected in H. Recently, in [12], Chill, Hauer and Kennedy extended results of [3], [4] by Arendt and Ter Elst to a nonlinear framework of j-elliptic functions ϕ : V → (−∞, +∞] generating a quasi maximal monotone operator ∂ j ϕ on H, where j : V → H is just a linear operator which is not necessarily injective. This enabled the authors of [12] to show that several coupled parabolicelliptic systems can be realized as a gradient system in a Hilbert space H and to extend the linear variational theory of the Dirichlet-to-Neumann operator to the nonlinear p-Laplace operator (see also [6,7,16] for further applications and extensions of this theory).…”

Section: Application To J-elliptic Functionsmentioning

confidence: 99%

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

Arendt

Hauer

2020

Pure Appl. Analysis

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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.

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