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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 \(\alpha<1\).

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