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The hypoelliptic Robin problem for quasilinear elliptic equations

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  • This paper is devoted to the study of a hypoelliptic Robin boundary value problem for quasilinear, second-order elliptic differential equations depending nonlinearly on the gradient. More precisely, we prove an existence and uniqueness theorem for the quasilinear hypoelliptic Robin problem in the framework of Hölder spaces under the quadratic gradient growth condition on the nonlinear term. The proof is based on the comparison principle for quasilinear problems and the Leray–Schauder fixed point theorem. Our result extends earlier theorems due to Nagumo, Akô and Schmitt to the hypoelliptic Robin case which includes as particular cases the Dirichlet, Neumann and regular Robin problems.

    Mathematics Subject Classification: Primary: 35J62; Secondary: 35H10, 35R25.


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  • Figure 1.  The unit outward normal $ \mathbf{n} $ and the conormal $ \boldsymbol\nu $ to $ \partial \Omega $

    Figure 2.  The open subset $ \Omega^{+} $ with boundary $ \partial \Omega^{+} $

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