Search results for "PDEs"
showing 10 items of 505 documents
A note on Serrin's overdetermined problem
2014
We consider the solution of the torsion problem $$−Δu = N \quad\mathrm{in}\quad Ω,\quad u = 0\quad\mathrm{on}\quad ∂Ω,$$ where Ω is a bounded domain in RN. ¶ Serrin's celebrated symmetry theorem states that, if the normal derivative uν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant Ct depending on t, where re and ri are the radii of an annulus containing ∂Ω and Γt is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly…
Uniqueness of positive radial solutions to singular critical growth quasilinear elliptic equations
2015
In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle \frac{\mu}{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+\lambda|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<\mu<((N-p)/p)^{p}$, $0\le s<p$ and $\lambda\in\mathbb{R}$. A related limiting problem is also considered.
A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows
2017
We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.
Removable singularities for div v=f in weighted Lebesgue spaces
2018
International audience; Let $w\in L^1_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p_{1/w}$, $1<p<+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.
Recovery of time-dependent coefficients from boundary data for hyperbolic equations
2019
We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.
On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
2014
In this article we prove Lp estimates for resolvents of Laplace–Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge (1987) in the Euclidean case and Shen (2001) for the torus. We follow Sogge (1988) and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain Lp Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig (1985); we illustrate the pertinence of Lp resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the constructi…
On critical behaviour in systems of Hamiltonian partial differential equations
2013
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
$n$-harmonic coordinates and the regularity of conformal mappings
2014
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
2016
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
The boundary Harnack inequality for infinity harmonic functions in Lipschitz domains satisfying the interior ball condition
2008
Abstract In this note, we give a short proof for the boundary Harnack inequality for infinity harmonic functions in a Lipschitz domain satisfying the interior ball condition. Our argument relies on the use of quasiminima and the notion of comparison with cones.