The Dirichlet problem without the maximum principle
Annales de l'Institut Fourier, to appear, 20 p.

Consider the Dirichlet problem with respect to an elliptic operator

A=-k,l=1dkakll-k=1dkbk+k=1dckk+c0

on a bounded Wiener regular open set Ω d , where a kl ,c k L (Ω,) and b k ,c 0 L (Ω,). Suppose that the associated operator on L 2 (Ω) with Dirichlet boundary conditions is invertible. Then we show that for all ϕC(Ω) there exists a unique uC(Ω ¯)H loc 1 (Ω) such that u| Ω =ϕ and Au=0.

In the case when Ω has a Lipschitz boundary and ϕC(Ω ¯)H 1/2 (Ω ¯), then we show that u coincides with the variational solution in H 1 (Ω).

Considérons le problème de Dirichlet par rapport à un opérateur elliptique

A=-k,l=1dkakll-k=1dkbk+k=1dckk+c0

sur un ensemble ouvert régulier de Wiener borné Ω d , où a kl ,c k L (Ω,) et b k ,c 0 L (Ω,). Supposons que 0 n’est pas une valeur propre de A avec conditions aux limites Dirichlet. Alors nous montrons que pour tout ϕC(Ω) il existe un unique uC(Ω ¯)H loc 1 (Ω) tel que u| Ω =ϕ et Au=0.

Dans le cas où Ω a une frontière Lipschitz et ϕC(Ω ¯)H 1/2 (Ω ¯), nous montrons que u coïncide avec la solution variationnelle dans H 1 (Ω).

Received : 2017-10-24
Revised : 2018-02-01
Accepted : 2018-03-13
Classification:  31C25,  35J05,  31B05
Keywords: Dirichlet problem, Wiener regular, holomorphic semigroup
@unpublished{AIF_0__0_0_A21_0,
     author = {Arendt, Wolfgang and ter Elst, A. F. M.},
     title = {The Dirichlet problem without the maximum principle},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Arendt, Wolfgang; ter Elst, A. F. M. The Dirichlet problem without the maximum principle. Annales de l'Institut Fourier, to appear, 20 p.

[1] Arendt, Wolfgang; Bénilan, Philippe Wiener regularity and heat semigroups on spaces of continuous functions, Topics in nonlinear analysis. The Herbert Amann anniversary volume, Birkhäuser (Progress in Nonlinear Differential Equations and their Applications) Tome 35 (1999), pp. 29-49 | Zbl 0920.35041

[2] Arendt, Wolfgang; Daners, Daniel The Dirichlet problem by variational methods, Bull. Lond. Math. Soc., Tome 40 (2008) no. 1, pp. 51-56 | Zbl 1167.35011

[3] Arendt, Wolfgang; Ter Elst, Antonius F. M. Gaussian estimates for second order elliptic operators with boundary conditions, J. Oper. Theory, Tome 38 (1997) no. 1, pp. 87-130 | Zbl 0879.35041

[4] Arendt, Wolfgang; Ter Elst, Antonius F. M. The Dirichlet-to-Neumann operator on C(Ω) (2019) (https://arxiv.org/abs/1707.05556, to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci.)

[5] Arendt, Wolfgang; Ter Elst, Antonius F. M.; Kennedy, James. B.; Sauter, Manfred The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., Tome 266 (2014) no. 3, pp. 1757-1786 | Zbl 1314.47062

[6] Arendt, Wolfgang; Nikolski, Nikolai Vector-valued holomorphic functions revisited, Math. Z., Tome 234 (2000) no. 4, pp. 777-805 | Zbl 0976.46030

[7] Dahlberg, Bjã¶Rn E. J. Estimates of harmonic measure, Arch. Rational Mech. Anal., Tome 65 (1977), pp. 275-288 | Zbl 0406.28009

[8] Daners, Daniel Heat kernel estimates for operators with boundary conditions, Math. Nachr., Tome 217 (2000), pp. 13-41 | Zbl 0973.35087

[9] Giaquinta, Mariano; Martinazzi, Luca An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Edizioni Della Normale (2005), ix+302 pages | Zbl 1093.35001

[10] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Springer, Grundlehren der Mathematischen Wissenschaften (1983) no. 224, xiii+513 pages | Zbl 0562.35001

[11] Littman, Walter; Stampacchia, Guido; Weinberger, Hans F. Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser., Tome 17 (1963), pp. 43-77 | Zbl 0116.30302

[12] Ouhabaz, El Maati Analysis of heat equations on domains, Princeton University Press, London Mathematical Society Monographs Series, Tome 31 (2005), xi+284 pages | Zbl 1082.35003

[13] Stampacchia, Guido Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, Tome 15 (1965), pp. 189-258 | Zbl 0151.15401