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$${\mathcal L}^{2,\lambda{}}$$ theory for nonlinear non-variational differential systems. (English) Zbl 0777.35028
Summary: In a bounded open set $$\Omega$$ consider the nonlinear system $a(x,u,Du,H(u))=b(x,u,Du),\tag{*}$ where $$a(x,u,p,\xi)$$ and $$b(x,u,p)$$ are vectors in $$\mathbb{R}^ N$$, $$N\geq 1$$, measurable in $$x$$ and continuous in the other variables, moreover, $$b(x,u,p)$$ has a linear growth and $$\xi\to a(x,u,p,\xi)$$ satisfies a further condition. We prove that $$\exists r_ 0>0$$ such that $$\forall B(r)\subset\Omega$$, with $$r\leq r_ 0$$, the Dirichlet problem $u\in H^ 2\cap H^ 1_ 0,\quad a(x,u,Du,H(u))=b(x,u,Du)\quad\text{in }B(r)$ has at least one solution. We further show that if $$u\in H^ 2(\Omega)$$ is a solution in $$\Omega$$ then: $$Du$$ is Hölder continuous in $$\Omega$$ if $$n=2$$, $$u$$ is Hölder continuous in $$\Omega$$ if $$n\leq 4$$.
Finally, we prove that if $$u\in H^ 2(\Omega)$$ is a solution of the system (*) and the vector $$a(x,u,p,\xi)$$ is of class $$C^ 1$$ in $$\xi$$ and satisfies certain continuity conditions in $$u$$ and in $$\partial a/\partial \xi$$, then, for any $$n$$, the vector $$Du$$ is partially Hölder continuous in $$\Omega$$ for every exponent $$\alpha<1$$.

##### MSC:
 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
##### Keywords:
existence; Hölder continuous solution; Dirichlet problem