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A class of De Giorgi type and Hölder continuity. (English) Zbl 0927.46022

Investigating the regularity of solutions of quasilinear elliptic equations with the standard \(m\)-growth condition one can use the function class \({\mathcal B}_m\), which was introduced by Ladyzhenskaya and Ural’tseva. The authors are interested in \(m(x)\)-growth conditions with \(m(x)\) being a positive, continuous, bounded function. The last conditions are of the form \[ c_1| z|^{m(x)}- c_0\leq F(x,z)\leq c_2| z|^{m(x)}- c_0, \] with \(F: \Omega\times \mathbb{R}^n\to \mathbb{R}\), \(\Omega\subset \mathbb{R}^n\), and \(1< p\leq m(x)\leq q\) for \(x\in\Omega\).
To study the regularity of the corresponding quasilinear equations, they introduce a class \({\mathcal B}_{m(x)}\) which is a generalization of the class \({\mathcal B}_m\). It is proved that if the function \(m(x)\) satisfies some additional conditions (in particular \(m(x)\) is Hölder continuous) then the functions belonging to \({\mathcal B}_{m(x)}\) are Hölder continuous. In consequence the Hölder continuity of minimizers of variational functionals satisfying \(m(x)\) growth conditions follows. As application the Hölder regularity of solutions of quasilinear equations with principal part in divergence form and with the same growth conditions can be proved.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49N60 Regularity of solutions in optimal control
47F05 General theory of partial differential operators
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