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Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic operator in divergence form (in particular I am interested in the case of the capillary equation, i.e. of prescribed mean curvature) without any zero order terms.

Is there any general existence result for classical solutions of the following Dirichlet problem? $$ \begin{cases} L u = 0 \qquad \text{in } \Omega;\\ u|_{\partial \Omega} = \phi. \end{cases} $$

I know that the answer is positive in the case of the minimal surface equation, but I was wonder if the result is still true even in the presence of a (nonlinear) first order term. Any help would be very appreciated!

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Yes, you can include a nonlinear first-order term, under certain conditions. See Elliptic Partial Differential Equations by Han & Lin, Section 6.6 (in the 2nd edition).

Let $L$ be of the form $$Lu = a_{ij}(x,u,\nabla u) u_{x_ix_j} + b(x,u,\nabla u),$$ with $a_{ij}$ uniformly elliptic and $a$ and $b$ Holder continuous. Using Leray-Schauder, they show that if a priori $C^{1,\beta}$ estimates hold for the modified problem $L_\sigma u = 0$ with $u|_{\partial\Omega} = \sigma \phi$, where $$L_\sigma u = a_{ij}(x,u,\nabla u) u_{x_ix_j} + \sigma b(x,u,\nabla u),$$ with $\sigma\in [0,1]$, and the estimates are uniform in $\sigma$, then a solution to the original problem $Lu =0$, $u|_{\partial \Omega} = \phi$ exists in $C^{2,\alpha}$. This is Theorem 6.23 in Han-Lin. They then show how this theorem applies to the minimal surface equation (using De Giorgi-Moser to get the needed estimates), where $b = 0$ so you don't have to worry about $\sigma$. The proof for nonzero $b$ should follow a similar outline.

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  • $\begingroup$ Hey! Thanks a lot for the answer! I've checked the reference and unfortunately they assume some extra regularity.. Namely they assume $\Omega$ to be $C^{2, \alpha}$ and moreover they assume $\phi \in C^{2, \alpha}$, while I'd like to allow corners in $\partial \Omega$.. $\endgroup$ – Onil90 Jul 24 '18 at 8:12
  • $\begingroup$ @Onil90 Good point. I don't know of any reference for the case of rough boundary/data. Perhaps one could try to apply Leray-Schauder in a lower regularity class (since the global estimates are limited by $\phi$ and $\partial \Omega$) but it may be tricky to show the a priori estimates hold for your equation in this case. $\endgroup$ – user126920 Jul 24 '18 at 15:16
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Prescribed mean curvature (of the graph of $u$) would be$$\nabla\cdot\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=c$$For an elliptic domain with $\phi$ linear in the direction of the smaller axis, the graph of $\phi$ may be a circle, in this case the solution should be spherical, but for some values of $c$ this spherical section will not be the graph of a function defined on $\Omega$ (its projection will include $\Omega$ strictly).

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  • $\begingroup$ Thanks for the answer! But I am interested in the case where $c$ actually depends also on $\nabla u$. $\endgroup$ – Onil90 Jul 24 '18 at 8:13
  • $\begingroup$ But if there were a general existence theorem, wouldn't it apply to the simple example I gave? Stanley Nelson's claim that "the proof for nonzero $b$ should follow a similar outline" seems doubtful to me, to say the least. $\endgroup$ – Jean Duchon Jul 24 '18 at 13:59

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