Let \(\Omega \subset {\mathbb{R}}^ n\) be a domain with \(C^{1,\beta}\) boundary (with \(\beta\in (0,1])\), \(Q_ T=\Omega \times (0,T)\), \(SQ=\partial \Omega \times (0,T)\) (with \(T>0).\)
The author considers the solutions u of the following boundary problem:
(*) \(u_ t=div A(X,u,Du)+B(X,u,Du)\) in \(Q_ T,\)
(**) \(A(X,u,Du)\cdot \gamma (x)+\psi (X,u)=0\) on SQ with \(u\in L(Q_ T)\) and
(***) \(Du\in L^{\infty}(\Omega \times (\epsilon,T))\) for any \(\epsilon\in (0,T).\)
Here A is a vector-valued function on \(Q_ T\times {\mathbb{R}}\times {\mathbb{R}}^ n\) satisfying \[ | A(x,t,z,p)-A(y,t,w,p)| \leq (1+| p|)^{m+1}[\Lambda | x-y|^{\beta}+\Lambda_ 1| z- w|^{\beta}], \] for any x,y\(\in \Omega\), \(0<t<T\), \(z,w\in {\mathbb{R}}\) and \(p\in {\mathbb{R}}^ n\) and \(\Lambda\), \(\Lambda_ 1\) are some positive constants, and \[ | A(x,t,z,p)-A(x,s,z,p)| \leq \Lambda (1+p)^{m+1}| t-s|^{\beta /m+2} \] for \(x\in \partial \Omega\), \(0<s<t<T\), and whose derivatives satisfy also some evaluations, B is a scalar-valued function with some natural restriction, and \(\psi\) is a scalar function such that \[ | \psi (X,z)-\psi (Y,w)| <\Psi (| x-y|^{\beta}+| t-s|)^{\beta /m+2}| +\Lambda_ 1| z-w|^{\beta}, \] where \(\Psi\) is a positive constant. Also \(\gamma(x)\) devotes the inner unit normal of point x and m is greater than -1. The main result is the following:
If u is a weak solution of (*)-(**)-(***), for \(\Lambda_ 1\) sufficiently small there exist a constant \(\sigma\in (0,\beta]\) such that for any \(\epsilon\in (0,T)\) and \(X,Y\in \Omega \times (\epsilon,T)\) \[ | Du(X)-Du(Y)| <C(| x-y|^{\sigma}+| t- s|^{\sigma /n+2}). \] The dependence of C and \(\beta\) is also stated.
The proof is first made in a simpler case, inspired by E. Di Benedetto and A. Friedman [J. Reine Angew. Math. 357, 1-22 (1985; Zbl 0549.35061)]. But some differences improve the results: for instance one works directly in cylinders with the appropriate scaling for the equations, and the proof applies regardless of the sign of m. Then the general case is proved from the special case by a simple variant of a perturbation argument of M. Giaquinta and E. Giusti [ibid. 351, 55-65 (1984; Zbl 0528.35014)], as modified to parabolic co-normal problems by the author [Ann. Mat. Pura Appl., IV. Ser. 148, 77-99, 397- 398 (1987; Zbl 0658.35050 resp. Zbl 0658.35051)]. The work ends with an existence theorem for the boundary value problem: \[ v_ t=div A(Dv)\text{ in } Q^+(R),\quad A^ n(Dv)+u=0\text{ on } Q^ 0(R), \] v\(=u\) on \(Q^*(R)\) in a suitable space for continuous u, with slightly additional hypothesis than in the previous result (here \(Q^+(R)\) and \(Q^ 0(R)\) are the scaled cylinders \(\{Y=(y,s):| X-Y| <R\), \(s<t_ 0\), \(y_ n>0\}\), respectively \(\{Y=(y,s):| X-Y| <R\), \(s<t_ 0\), \(y_ n=0\}\), \(Q^*(R)\) is the intersection of the “parabolic” parts of \(Q^+(R)\) and \(Q^ 0(R)\).