On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. (English) Zbl 0704.35024

Campanato, S.

Ann. Mat. Pura Appl., IV. Ser. 137, 83-122 (1984).

The author considers second-order nonlinear parabolic systems of the type \[ -\sum^{n}_{i=1}D_ iA^ i(X,u,Du)+\frac{\partial u}{\partial t}=- \sum^{n}_{i=1}D_ iB^ i(X,u)+B^ 0(X,u,Du) \] in the cylinder \(Q=\Omega \times (-T,0)\), whose generic point is \(X=(x,t)\). He supposes that the system is strongly parabolic and that the vectors \(A^ i,B^ i,B^ 0\) have strictly controlled growths. Under some additional hypotheses on \(A^ i\), he proves the main result: When \(n\leq 2\) the solutions are \(\mu\)-partial Hölder continuous in Q for a suitable \(\mu\in (0,1)\) and their singular sets have zero n-dimensional Hausdorff measure; whereas, for any n, the solutions are \(\mu\)-partial Hölder continuous for all \(\mu\in (0,1)\), but, about their singular set \(Q_ 0\) the author can merely say, at present, that the Lebesgue measure of \(Q_ 0\) is zero.
These results are obtained as a particular case of regularity or partial regularity in the \({\mathcal L}^{2,\lambda}(Q,d)\) spaces. It is remarkable that in the theory of the \({\mathcal L}^{2,\lambda}\)-regularity for nonlinear parabolic systems, as developed by the author during the last few years, those systems of the type \(-\sum^{n}_{i=1}D_ iA^ i(Du)+\partial u/\partial t=0\) play an analogous role to that played by linear systems, with constant coefficients and reduced to the principal part, in the theory of linear or quasilinear systems. These systems are intensively studied in the paper, and the results obtained are of value by themselves. Moreover, other results are obtained when the coefficients \(A^ i\) are of the type \(A^ i(X,Du)\).

MSC:

35D10	Regularity of generalized solutions of PDE (MSC2000)
35K55	Nonlinear parabolic equations
35K25	Higher-order parabolic equations

Keywords:

strongly parabolic; partial Hölder continuous; singular sets; Hausdorff measure

References:

[1]	S.Campanato,Sistemi ellittici in forma divergenza. Regolarità all’interno, « Quaderni » Ann. Scuola Norm. Sup. Pisa, 1980. · Zbl 0453.35026
[2]	S.Campanato,Hölder continuity of the solutions of some nonlinear elliptic systems, Advances in Math.,42 (1983). · Zbl 0519.35027
[3]	S.Campanato,Hölder continuity and partial Hölder continuity results for H^1,q-solutions of non-linear elliptic systems with controlled growth, to appear in Rend. di Milano. · Zbl 0576.35041
[4]	S.Campanato,L^p-regularity for weak solutions of parabolic systems, Ann. Scuola Norm. Sup. Pisa,7 (1980). · Zbl 0456.35009
[5]	S.Campanato,Partial Hölder continuity of solutions of quasi-linear parabolic systems of second order with linear growth, Rend. Sem. Mat. Univ. Padova,64 (1981). · Zbl 0496.35012
[6]	S.Campanato,L^p-regularity and partial Hölder continuity for solutions of second order parabolic systems with strictly controlled growth, Ann. Mat. Pura Appl.,182 (1980). · Zbl 0497.35014
[7]	S.Campanato, Equazioni paraboliche del seconda ordine e spaziL^2, θ(Ω, δ), Ann. Mat. Pura Appl.,73 (1966). · Zbl 0144.14101
[8]	P.Cannarsa,Second order non variational parabolic systems, Boll. Un. Mat. Ital., Analisi Funz. e Appl.,82, C.N. 1 (1981). · Zbl 0473.35043
[9]	G.Da Prato, SpaziL^p, θ(Ω, δ) e loro proprietá, Ann. Mat. Pura Appl.,69 (1965).
[10]	M.Giaquinta - E.Giusti,Partial regularity for the solutions to non-linear parabolic systems, Ann. Mat. Pura Appl.,97 (1973). · Zbl 0276.35062
[11]	O. A.Ladyzenskaja - V. A.Solonnikov - N. N.Ural’ceva,Linear and quasilinear equations of parabolic type, Amer. Mathem. Soc. Translations of Mathem. Monographs, 1968.
[12]	J. L.Lions,Equations differentielles operationnelles, Springer, 1961. · Zbl 0098.31101
[13]	J. L.Lions - E.Magenes,Problèmes aux limites non homogènes et applications, Dunod, 1968. · Zbl 0165.10801
[14]	M.Marino - A.Maugeri,L^ptheory and partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth, to appear. · Zbl 0576.35063
[15]	A.Maugeri,Partial Hölder continuity for the derivatives of order (m − 1)of solutions to 2 m order quasilinear parabolic systems with linear growth, to appear. · Zbl 0518.35049