zbMATH — the first resource for mathematics

On parabolic systems with variational structure. (English) Zbl 0593.35054
The author deals with the parabolic system of the second order for a vector function $$u=(u^ 1,...,u^ N)$$, $$N>1$$, $u^ i_ t=\sum_{\alpha,\beta}D_{\alpha}(a_{\alpha,\beta}(x,u)D_{\beta}u^ i)+a^ i_ 0(x,u\quad,\nabla u),\quad i=1,...,N,$ in the cylinder $$Q=\Omega \times (0,T)$$, where $$\Omega \subset R^ n$$ is a bounded Lipschitz domain. The functions $$a_{\alpha,\beta}$$, $$a^ i_ 0$$ are Lipschitz, $$a_{\alpha,\beta}$$-bounded, $$a^ i_ 0(x,u,p)$$ having quadratic growth in p. Uniform ellipticity in second order term and the variational structure of the system is assumed. Moreover a one-sided condition is imposed: $$a_ 0(x,u(x),\nabla u(x)):$$ $$u\geq -\mu | \nabla u(x)|^ 2-K$$, for each $$u\in L_{\infty}(\Omega)\cap W^ 1_ 2(\Omega)$$ and a.e. $$x\in \Omega$$ $$(\mu,K>0)$$. Employing the method of continuation the existence of a solution is considered for the Cauchy- Dirichlet problem $u(x,0)=u_ 0(x),\quad x\in \Omega;\quad u(x,t)=g(x),\quad x\in \partial \Omega,$ with g small in $$L_{\infty}(\partial \Omega)$$-norm. The main result yields (under some regularity conditions on $$u_ 0,g)$$ a solution-Hölder continuous in $$\bar Q,$$ for the case $$n=2$$. For general $$n\geq 2$$ the problem is reduced to a certain condition which is proved to hold for $$n=2$$.
Reviewer: P.Polacik

MSC:
 35K55 Nonlinear parabolic equations 35K45 Initial value problems for second-order parabolic systems 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
Full Text:
References:
 [1] ARONSON, D.G.: Non-negative solutions of linear parabolic equations; Ann. Sc. Norm. Sup. Pisa,22(1968), 607-694 · Zbl 0182.13802 [2] EELLS, J., J.H. SAMPSON: Harmonic mappings of Riemannian manifolds; Am. J. Math.86(1964), 109-160 · Zbl 0122.40102 [3] FREHSE, J.: On Two-Dimensional Quasi-Linear Elliptic Systems, manusc. math.28(1979), 21-49 · Zbl 0415.35025 [4] GIAQUINTA, M., M. STRUWE: An optimal regularity result for a class of quasilinear parabolic systems, manusc. math.36(1981), 223-239 · Zbl 0475.35026 [5] GILBARG, D., N.S. TRUDINGER: Elliptic Partial Differential Equations of Second Order, Springer Grundlehren 224, Berlin, Heidelberg, New York, Tokyo, 2nd edition, 1983 · Zbl 0562.35001 [6] HAMILTON, R.S.: Harmonic Maps of Manifolds with Boundary, Springer Lecture Notes 471, Berlin, Heidelberg, New York, 1975 · Zbl 0308.35003 [7] HILDEBRANDT, S., H. KAUL, K.-O. WIDMAN: An Existence Theorem for Harmonic Mappings of Riemannian Manifolds, Acta Math.138 (1977), 1-16 · Zbl 0356.53015 [8] JOST, J.: Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses; manusc. math.34(1981), 17-25 · Zbl 0459.58013 [9] LADYSHENSKAYA, O.A., V.A. SOLONNIKOV, N.N. URAL’CEVA: Linear and Quasilinear Equations of Parabolic Type; Transl. Math. Monogr. 23, AMS, Providence, 1968 [10] LIONS, J.L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969 [11] MORREY, C.B. Jr.: Multiple Integrals in the Calculus of Variations, Springer, New York, 1966 · Zbl 0142.38701 [12] STRUWE, M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems; manusc. math.35(1981), 125-145 · Zbl 0519.35007 [13] STRUWE, M.: Some regularity results for quasilinear parabolic systems, to appear in: Commentat. Math. Univ. Carol. [14] STRUWE, M.: On the evolution of harmonic mappings, to appear in: Comm. Math. Helv. · Zbl 0595.58013 [15] TOLKSDORF, P.: On some parabolic variational problems with quadratic growth, to appear in: Ann. Sc. Norm. Sup. Pisa [16] WAHL, W. VON: Verhalten der Lösungen parabolischer Gleichungen für t?? mit Lösbarkeit im Großen; Nachr. Akad. Wiss. Göttingen5(1981) · Zbl 0492.35035 [17] WIEGNER, M.: On two-dimensional elliptic systems with a one-sided condition, Math. Z.178(1981), 493-500 · Zbl 0474.35046
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.