Abstract
Let \({\Omega\subset\mathbb{R}^N}\) be an arbitrary open set with boundary \({\partial \Omega, 1 < p < \infty}\) and let \({f\in L^q(\Omega)}\) for some q > N > 1. In the first part of the article, we show that weak solutions of the quasi-linear elliptic equation \({-{\rm div}(|\nabla u|^{p-2} \nabla u)+a(x)|u|^{p-2}u=f}\) in Ω with the nonlocal Robin type boundary conditions formally given by \({|\nabla u|^{p-2} \partial u/\partial\nu+b(x)|u|^{p-2}u+\Theta_p(u)=0}\) on \({\partial \Omega}\) belong to L ∞(Ω). In the second part, assuming that Ω has a finite measure, we prove that for every \({p \in (1,\infty)}\), a realization of the operator \({\Delta_p}\) in L 2(Ω) with the above-mentioned nonlocal Robin boundary conditions generates a nonlinear order-preserving semigroup \({(S_\Theta(t))_{t \ge 0}}\) of contraction operators in L 2(Ω) if and only if \({\partial \Omega}\) is admissible (in the sense of the relative capacity) with respect to the (N − 1)-dimensional Hausdorff measure \({\fancyscript{H}^{N-1}|_{\partial \Omega}}\). We also show that this semigroup is ultracontractive in the sense that, for every \({u_0 \in L^q(\Omega)}\) (q ≥ 2) one has \({S_\Theta(t)u_0 \in L^\infty(\Omega)}\) for every t > 0. Moreover, \({\|S_\Theta(t)}\) satisfies the following (L q − L ∞)-Hölder type estimate: there is a constant C ≥ 0 such that for every t > 0 and \({u_0, v_0 \in L^q(\Omega)}\) (q ≥ 2),
where β, δ, and γ are explicit constants depending on N, p, and q only.
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References
Arendt W., Warma M.: The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19, 341–363 (2003)
Arendt W., Warma M.: Dirichlet and Neumann boundary conditions: what is in between?. J. Evol. Equ. 3, 119–135 (2003)
Barthélemy L.: Invariance d’un convex fermé par un semi-groupe aassocié à une forme non-linéaire. Abstr. Appl. Anal. 1, 237–262 (1996)
Biegert, M.: Elliptic problems on varying domains. Ph.D Dissertation, University of Ulm (2005)
Biegert M.: A priori estimates for the difference of solutions to quasi-linear elliptic equations. Manuscripta Math. 133, 273–306 (2010)
Biegert M., Warma M.: The heat equation with nonlinear generalized Robin boundary conditions. J. Differ. Equ. 247, 1949–1979 (2009)
Biegert M., Warma M.: Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad” domains. Adv. Differ. Equ. 15, 893–924 (2010)
Chill, R., Warma, M.: Dirichlet and Neumann boundary conditions for the p-Laplace operator: what is in between? In: Proceedings of Royal Society Edinburgh Section A 142 (2012, to appear)
Cipriani F., Grillo G.: Uniform bounds for solutions to quasilinear parabolic equations. J. Differ. Equ. 177, 209–234 (2001)
Cipriani F., Grillo G.: L q−L ∞ Hölder continuity for quasilinear parabolic equations associated to Sobolev derivations. J. Math. Anal. Appl. 270, 267–290 (2002)
Cipriani F., Grillo G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and application to minimal surfaces. J. Reine. Angew. Math. 562, 201–235 (2003)
Daners D.: Robin boundary value problems on arbitrary domains. Trans. Am. Math. Soc. 352, 4207–4236 (2000)
Daners D.: A priori estimates for solutions to elliptic equations on non-smooth domains. Proc. R. Soc. Edinburgh Sect. A 132, 793–813 (2002)
Daners D., Drábek P.: A priori estimates for a class of quasi-linear elliptic equations. Trans. Am. Math. Soc. 361, 6475–6500 (2009)
Davies E.B.: Heat Kernel and Spectral Theory. Cambridge University Press, Cambridge (1989)
DiBenedetto E.: Degenerate Parabolic Equations. Springer, New York (1993)
Drábek P., Milota J.: Methods of Nonlinear Analysis. Applications to Differential Equations. Birkhäuser Advanced Texts, Birkhäuser, Basel (2007)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Federbush P.: Partially alternate derivative of a result of Nelson. J. Math. Phys. 10, 50 (1969). doi:10.1063/1.1664760
Gesztesy F., Mitrea M.: Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities. J. Differ. Equ. 247, 2871–2896 (2009)
Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)
Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. Lecture Notes in Mathematics, vol. 1563. Springer, Berlin (1993)
Hajłasz P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5, 403–415 (1996)
Hajłasz P., Koskela P., Tuominen H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254, 1217–1234 (2008)
Maz’ya V.G.: Sobolev Spaces. Springer, Berlin (1985)
Maz’ya V.G., Poborchi S.V.: Differentiable Functions on Bad Domains. World Scientific Publishing, Singapore (1997)
Minty G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Minty G.J.: On the solvability of nonlinear functional equations of monotonic type. Pacific J. Math. 14, 249–255 (1964)
Murthy M.K.V., Stampacchia G.: Boundary value problems for some degenerate-elliptic operators. Ann. Mat. Pura Appl. 80, 1–122 (1968)
Showalter R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence (1997)
Shvartsman P.: On extensions of Sobolev functions defined on regular subsets of metric measure spaces. J. Approx. Theory 144, 139–161 (2007)
Velez-Santiago A.: Quasi-linear boundary value problems with generalized nonlocal boundary conditions. Nonlinear Anal. 74, 4601–4621 (2011)
Velez-Santiago A.: Solvability of linear local and nonlocal Robin problems over \({C(\overline{\Omega})}\). J. Math. Anal. Appl. 386, 677–698 (2012)
Velez-Santiago A., Warma M.: A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions. J. Math. Anal. Appl. 372, 120–139 (2010)
Warma, M.: The Laplacian with General Robin Boundary Conditions. Ph.D Dissertation, University of Ulm (2002)
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Warma, M. The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets. Annali di Matematica 193, 203–235 (2014). https://doi.org/10.1007/s10231-012-0273-y
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DOI: https://doi.org/10.1007/s10231-012-0273-y
Keywords
- The p-Laplace operator
- Nonlocal Robin boundary conditions on arbitrary open sets
- Bounded weak solutions
- A priori estimates
- Relative capacity
- Nonlinear ultracontractive semigroups
- Hölder type estimates