Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.
Let $n\ge3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctan\lambda_i(D^2u)=\theta(x) \] under the critical and supercritical phase condition $\theta\ge (n-2)\pi/2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.
We establish the local boundedness of time derivatives of bounded viscosity solutions to the normalized parabolic $p$-Laplace equation for every $p\in(1,\infty)$. The proof combines a logarithmic Bernstein argument with quantitative stability to derive a recursive estimate between different regularizations. For $p=1$ a...
For \(\frac12\le s<1\), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and \(L^2\)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass \(a_*\). For \(0<a\le a_*\),...
We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. Th...
Let $(M,g,J)$ be a closed K\"ahler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler--Lagrange equations associated with the Beckner--Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity...
We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carath\'eodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in \(L^1\). O...
Pierre Bousquet, C. Mariconda, G. Treu· 1 citation
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