By Gabriella Caristi, Enzo Mitidieri (auth.), Erik Koelink, Jan van Neerven, Ben de Pagter, Guido Sweers, Annemarie Luger, Harald Woracek (eds.)
Capturing the cutting-edge of the interaction among partial differential equations, practical research, maximal regularity, and chance conception, this quantity was once initiated on the Delft convention at the social gathering of the retirement of Philippe Clément. will probably be of curiosity to researchers in PDEs and practical analysis.
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Extra info for Partial Differential Equations and Functional Analysis: The Philippe Clément Festschrift
Serrin, H. , 189 (2002), 79–142. G. Simader, An elementary proof of Harnack’s inequality for Schr¨ odinger operators and related topics, Math. Z. 203 (1990), 129–152. G. Simader, Mean value formula, Weyl’s lemma and Liouville theorems for ∆2 and Stoke’s System, Results Math. 22 (1992), 761–780.  R. Soranzo, Isolated Singularities of Positive Solutions of a Superlinear Biharmonic Equation, Potential Analysis, 6 (1997), 57–85.  G. Stampacchia, Le probl`eme de Dirichlet pour les ´equations elliptiques du second ordre a ` coeﬃcients discontinus, Ann.
28 C. Carstensen Indeed, many second-order elliptic boundary value problems are recast in a weak form a(u, v) = b(v) for all v ∈ V, where (V, a) is some Hilbert space with induced norm · a and b ∈ V ∗ has the Riesz representation u ∈ V . Given a subspace Vh , the discrete solution uh ∈ Vh satisﬁes a(uh , vh ) = b(vh ) for all vh ∈ Vh . The error e := u − uh ∈ V then satisﬁes the best-approximation property e a = min vh ∈Vh u − vh a. ) The classical estimation of the upper bound u − vh a replaces vh by the nodal interpolation operator applied to the exact solution u.
Gidas, J. Spruck, Global and Local behavior of Positive Solutions of Nonlinear Elliptic Equations, Comm. Pure Appl. , 34 (1981), 525–598.  D. Gilbarg, N. Trudinger, Elliptic partial diﬀerential equations of second order, Springer Verlag, 1983. -Ch. Grunau, G. Ann. 307 (1997), 589–626. 26 G. Caristi and E. M. Hinz, H. Kalf, Subsolution estimates and Harnack inequality for Schr¨ odinger operators, J. Reine Angew. , 404 (1990), 118–134.  H. Mˆ aagli, F. Toumi, & M. Zribi, Existence of positive solutions for some polyharmonic nonlinear boundary-value problems, Electronic Journal of Diﬀerential Equations, Vol.