Related papers: Schauder Estimates for Germs by Scaling
We extend the semigroup approach used in [23,21] to provide alternative proofs of the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions. As an application of them, we construct the local-in-time…
In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in $\mathbb R^2$. Then we establish an uniform estimate for…
We prove unique continuation properties for solutions of the evolution Schr\"odinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up…
In this paper, we first establish decay estimates for the fractional and higher-order fractional H\'enon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the…
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first…
The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external…
We prove a unique continuation principle or uncertainty relation valid for Schr\"odinger operator eigenfunctions, or more generally solutions of a Schr\"odinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution…
We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method.…
We are concerned with wave equations associated to some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated to…
In this paper we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation $-\Delta u=e^u~~~{in }~~~\Omega$, where $\Omega$ is an open set of $R^3$. By improving the partial regularity…
Many applications, such as systems of interacting particles in physics, require the simulation of diffusion processes with singular coefficients. Standard Euler schemes are then not convergent, and theoretical guarantees in this situation…
In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical…
We studied the Benamou--Brenier formulation of the Schr\"odinger problem, focusing on a gap between theoretical results and applications, that often involve measures with unbounded support. While the existing proof in the literature relies…
The Schauder estimates are among the oldest and most useful tools in the modern theory of elliptic partial differential equations (PDEs). Their influence may be felt in practically all applications of the theory of elliptic boundary-value…
We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schr\"odinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for…
We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation $u_t= \Delta u+V(x)f(u)$, we rule out the…
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up…
We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $\chi_{(-\infty,E]}(H_L)$ of a Schr\"odinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential.…
Real blow-ups and more refined "zooms" play a key role in the analysis of singularities of complex-analytic differential modules. They do not change the underlying topology, but the uniform structure. This suggests to revisit the cohomology…
We propose a new scaling theory for general quantum breakdown phenomena. We show, taking Landau-Zener type breakdown as a particular example, that the breakdown phenomena can be viewed as a quantum phase transition for which the scaling…