×

Contractive properties for the heat equation in Sobolev spaces. (English) Zbl 0577.46037

Let S(t) denote the heat semigroup with zero Dirichlet data, acting on functions in a bounded smooth domain \(\Omega\). For \(1<p<\infty\), let \(q=p/(p-1)\), and let \(\lambda_ p\) be the smallest eigenvalue of - 4/pq\(\cdot \Delta\) with the boundary condition \[ 4/pq\cdot \partial_{\nu}u+(n-1)\cdot H\cdot u=0, \] where H is the mean curvature of \(\partial \Omega\). It is shown that \(e^{\lambda_ pt}\cdot S(t)\) is a contraction in \(W_ 0^{1,p}(\Omega)\). Thus, if \(H\geq 0\) everywhere on \(\partial \Omega\), then the well-known result follows that the gradient of \(u(\cdot,t)=S(t)u_ 0\) attain its maximum for \(t=0\), by sending p to infinity. A related result is shown for the case of zero Neumann boundary conditions. For the case of arbitrary geometries, equivalent norms to the \(W^{1,p}\)-norm are constructed explicitly for which S(t) is a contraction semigroup.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K05 Heat equation
47D03 Groups and semigroups of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alikakos, N. D.; Evans, L. C., Continuity of the gradient of the solution of certain degenerate parabolic equations, J. Math. Pures Appl., 62, 253-268 (1983) · Zbl 0529.35039
[2] Alikakos, N. D.; Rostamian, R., Gradient estimates for degenerate diffusion equations, I, Math. Ann., 259, 53-70 (1982) · Zbl 0465.35051
[3] Bergh, J.; Löfström, J., Interpolation Spaces (1976), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0344.46071
[4] Bernstein, S., Sur la nature analytique des solutions de certaines équatons aux dérivées partielles du second order, Math. Ann., 59, 20-76 (1904) · JFM 35.0354.01
[5] Courant, R.; Hilbert, D., Methoden der mathematischen Physik (1931), Springer-Verlag: Springer-Verlag Berlin · JFM 63.0449.05
[6] DiBenedetto, E.; Friedman, A., Regularity of solutions of nonlinear parabolic systems, J. Reine Angew. Math., 349, 83-128 (1984) · Zbl 0527.35038
[7] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0144.34903
[8] Ladyzenskaya, O. A.; Uraltseva, N. N.; Solonnikov, V. A., Linear and Quasilinear Equations of Parabolic Type (1968), Amer. Math. Soc: Amer. Math. Soc Providence, R.I
[9] Lions, J. L.; Magenes, E., Problemi ai limiti non omogenei, III, Ann. Scuola Norm. Sup. Pisa, 15, 39-101 (1961) · Zbl 0115.31401
[10] Lions, P. L., Une ineqalité pour les opérateurs elliptiques du second ordre, Ann. Mat. Pura Appl., 127, 1-11 (1981), (4) · Zbl 0468.35040
[11] Londen, S. O.; Nohel, J. A., Nonlinear Volterra integrodifferential equation occurring in heat flow, J. Integral Equations, 6, 11-50 (1984) · Zbl 0537.45011
[12] Pierre, M., Uniqueness of the solutions of \(u_t\) − \( Δφ (u) = 0\) with initial datum a measure, Nonlinear Anal. TMA, 6, 175-187 (1982) · Zbl 0484.35044
[13] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0153.13602
[14] Serrin, J. B., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A, 264, 413-496 (1969) · Zbl 0181.38003
[15] Sperb, R., Maximum Principles and Their Applications (1981), Academic Press: Academic Press New York · Zbl 0454.35001
[16] Spivak, M., A Comprehensive Introduction to Differential Geometry, I-V (1975), Publish or Perish: Publish or Perish Boston
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.