Dirichlet regularity and degenerate diffusion. (English) Zbl 1205.35139

Arendt, Wolfgang; Chovanec, Michal

Trans. Am. Math. Soc. 362, No. 11, 5861-5878 (2010).

The authors investigate the problem: when a natural realization of the operator \(m \Delta\) in \(C_0 (\Omega): = \{ u \in C (\bar{\Omega}) : u |_{\partial \Omega}= 0 \}\) generates a \(C_0\)-semigroup. Here \(\Omega \subset \mathbb{R}^{N}\) is an open bounded set and \(m : \Omega \to (0, \infty)\) is measurable and locally bounded.
If \(\Omega\) is Dirichlet regular, then the operator generates a positive contraction semigroup on \(C_0 (\Omega)\) whenever \(1/m \in L^p_{\text{loc}} (\Omega)\) for some \(p > N/2\). If \(m(x)\) does not go fast enough to \(0\) as \(x \to \partial \Omega\), then Dirichlet regularity is necessary. In the case where \(|m(x)| \leq c \cdot \text{dist} (x, \partial \Omega)^2\), the authors show that \(m_0 \Delta\) generates a semigroup on \(C_0 (\Omega)\) without any regularity assumptions on \(\Omega\). They show that the condition for degeneration of \(m\) near the boundary is optimal.

Reviewer: Miklavž Mastinšek (Maribor)

MSC:

35K05	Heat equation
47D06	One-parameter semigroups and linear evolution equations
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

Dirichlet regularity; degenerate diffusion; \(C_0\)-semigroups

References:

[1]	Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. · Zbl 0978.34001
[2]	Wolfgang Arendt and Philippe Bénilan, Wiener regularity and heat semigroups on spaces of continuous functions, Topics in nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 35, Birkhäuser, Basel, 1999, pp. 29 – 49. · Zbl 0920.35041
[3]	W. Arendt, Heat Kernels. Internetseminar 2005/2006, http://www.uni-ulm. de/mawi/iaa/ members/professors/arendt.html, 2005/06.
[4]	Wolfgang Arendt, Positive semigroups of kernel operators, Positivity 12 (2008), no. 1, 25 – 44. · Zbl 1151.47047
[5]	Wolfgang Arendt, Paul R. Chernoff, and Tosio Kato, A generalization of dissipativity and positive semigroups, J. Operator Theory 8 (1982), no. 1, 167 – 180. · Zbl 0494.47026
[6]	Wolfgang Arendt and Daniel Daners, The Dirichlet problem by variational methods, Bull. Lond. Math. Soc. 40 (2008), no. 1, 51 – 56. · Zbl 1167.35011
[7]	Wolfgang Arendt and Daniel Daners, Varying domains: stability of the Dirichlet and the Poisson problem, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 21 – 39. · Zbl 1155.35011
[8]	Markus Biegert and Mahamadi Warma, Regularity in capacity and the Dirichlet Laplacian, Potential Anal. 25 (2006), no. 3, 289 – 305. · Zbl 1198.35108
[9]	Piermarco Cannarsa, Genni Fragnelli, and Dario Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media 2 (2007), no. 4, 695 – 715. · Zbl 1140.93011
[10]	T. Cazenave, A. Haraux, Introduction aux problèmes d’évolution semi-linéaires elliptiques, Paris, 1990. · Zbl 0786.35070
[11]	E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. · Zbl 0699.35006
[12]	E. B. Davies, \?\textonesuperior properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417 – 436. · Zbl 0583.35032
[13]	R. Dautray, J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. I, Springer, Berlin, 1990. · Zbl 0784.73001
[14]	Michael Demuth and Jan A. van Casteren, Stochastic spectral theory for selfadjoint Feller operators, Probability and its Applications, Birkhäuser Verlag, Basel, 2000. A functional integration approach. · Zbl 0980.60005
[15]	Xuan Thinh Duong and El Maati Ouhabaz, Complex multiplicative perturbations of elliptic operators: heat kernel bounds and holomorphic functional calculus, Differential Integral Equations 12 (1999), no. 3, 395 – 418. · Zbl 1008.47020
[16]	K. D. Elworthy, Stochastic flows and the \?\(_{0}\)-diffusion property, Stochastics 6 (1981/82), no. 3-4, 233 – 238. · Zbl 0489.58037
[17]	William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468 – 519. · Zbl 0047.09303
[18]	Robert Haller-Dintelmann, Matthias Hieber, and Joachim Rehberg, Irreducibility and mixed boundary conditions, Positivity 12 (2008), no. 1, 83 – 91. · Zbl 1145.35061
[19]	Stefan Hildebrandt, On Dirichlet’s principle and Poincaré’s méthode de balayage, Math. Nachr. 278 (2005), no. 1-2, 141 – 144. · Zbl 1157.35340
[20]	N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. · Zbl 0253.31001
[21]	W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43 – 77. · Zbl 0116.30302
[22]	Gunter Lumer, Perturbation de générateurs infinitésimaux, du type ”changement de temps”, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 271 – 279 (French, with English summary). · Zbl 0263.47035
[23]	Gunter Lumer, Problème de Cauchy pour opérateurs locaux et ”changement de temps”, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xxiii, 409 – 446 (French, with English summary). Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday. · Zbl 0303.35007
[24]	Petr Mandl, Analytical treatment of one-dimensional Markov processes, Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York, 1968. · Zbl 0179.47802
[25]	Alan McIntosh and Andrea Nahmod, Heat kernel estimates and functional calculi of -\?\Delta , Math. Scand. 87 (2000), no. 2, 287 – 319. · Zbl 1069.35023
[26]	Keith Miller, Exceptional boundary points for the nondivergence equation which are regular for the Laplace equation and vice-versa, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 315 – 330. · Zbl 0164.13102
[27]	W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986.
[28]	El Maati Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. · Zbl 1082.35003
[29]	El-Maati Ouhabaz, Invariance of closed convex sets and domination criteria for semigroups, Potential Anal. 5 (1996), no. 6, 611 – 625. · Zbl 0868.47029
[30]	M. M. H. Pang, \?\textonesuperior properties of two classes of singular second order elliptic operators, J. London Math. Soc. (2) 38 (1988), no. 3, 525 – 543. · Zbl 0675.47054
[31]	Christian G. Simader, Equivalence of weak Dirichlet’s principle, the method of weak solutions and Perron’s method towards classical solutions of Dirichlet’s problem for harmonic functions, Math. Nachr. 279 (2006), no. 4, 415 – 430. · Zbl 1163.35003
[32]	Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[33]	J. van Casteren, Generators of Strongly Continuous Semigroups, Research Notes in Mathematics 115, Pitman, Boston, 1985. · Zbl 0576.47023