Uniform estimates for elliptic equations with Carathéodory nonlinearities at the interior and on the boundary
Edgar Antonio, Martín P. Árciga-Alejandre, Rosa Pardo, Jorge Sánchez Ortiz
TL;DR
This work addresses explicit uniform $L^{\infty}$ bounds for weak solutions of elliptic problems with simultaneous interior and boundary nonlinearities of slightly subcritical Carathéodory type. The authors combine a De Giorgi–Nash–Moser iterative scheme with elliptic regularity and the Gagliardo–Nirenberg interpolation to derive an explicit bound $h_m(\|u\|_{L^{\infty}(\Omega)}) \le C_{\varepsilon} a_M^{A+\varepsilon} (1+\|u\|_{H^1(\Omega)}^{(2^*_{N/r}-2)(A+\varepsilon)})$ for every $\varepsilon>0$, where $h_m$ merges interior and boundary nonlinearities via $h$ and $h_B$, and $a_M$ encodes data regularity. The result provides a priori control from $H^1$-bounds, enabling Leray–Schauder degree arguments for existence and extending known single-domain results to the coupled interior-boundary setting. The analysis delineates two parameter regimes and yields corollaries that bound solutions in terms of $L^{2^*}$ and $L^{2_*}$ norms, ultimately ensuring uniform $L^{\infty}$ control and compactness in $C(\overline{\Omega})$ for sequences of solutions. Overall, the paper advances the theory of elliptic problems with nonlinear boundary conditions by delivering fully explicit, regime-sensitive a priori estimates.
Abstract
We establish an explicit uniform a priori estimate for weak solutions to slightly subcritical elliptic problems with nonlinearities simultaneously at the interior and on the boundary. Our explicit $L^{\infty}(Ω)$ a priori estimates are in terms of powers of their $H^{1}(Ω)$ norms. To prove our result, we combine a De Giorgi-Nash-Moser's iteration scheme together with elliptic regularity and the Gagliardo-Nirenberg's interpolation inequality.