Showing papers in "Communications in Partial Differential Equations in 2013"
TL;DR: Weak solutions to parabolic integro-differential operators of order α ∈ (α 0, 2) are studied in this article, where local a priori estimates of Holder norms and a weak Harnack inequality are proved.
Abstract: Weak solutions to parabolic integro-differential operators of order α ∈ (α0, 2) are studied. Local a priori estimates of Holder norms and a weak Harnack inequality are proved. These results are rob...
138 citations
TL;DR: In this article, the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from...
Abstract: In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from ...
98 citations
TL;DR: In this article, the authors consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait and investigate the existence of travelling wave solutions.
Abstract: We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.
84 citations
TL;DR: In this paper, the authors extensively develop Carleman estimates for the wave equation and give some applications, such as exact controllability problem for wave equations with potentials, inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux, and constructive algorithm to rebuild the potential.
Abstract: In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.
81 citations
TL;DR: In this article, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Eucli...
Abstract: In [4] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Eucli...
74 citations
TL;DR: In this article, the authors prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel.
Abstract: We prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.
73 citations
TL;DR: In this paper, the authors consider the class of semi-stable positive solutions to semilinear equations in a bounded domain Ω ∈ n of double revolution, that is, a domain invariant under rotations of the firs.
Abstract: We consider the class of semi-stable positive solutions to semilinear equations − Δu = f(u) in a bounded domain Ω ⊂ ℝ n of double revolution, that is, a domain invariant under rotations of the firs...
66 citations
TL;DR: In this article, a unified approach to splitting theorems, symmetry results and overdetermined elliptic problems is proposed, based on the existence of a stable solution to the semilinear equation on a Riemannian manifold with non-negative Ricci curvature.
Abstract: Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation − Δu = f(u) on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary.
66 citations
TL;DR: In this article, the authors analyzed the global existence of classical solutions to the initial boundary value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons.
Abstract: In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled ...
61 citations
TL;DR: For the parabolic-elliptic Keller-Segel system with critical porous-medium diffusion in dimension ≥ 3, it is known that there is a critical value of the chemotactic sensitivity (measuring in some sense the strength of the drift term) above which there are solutions blowing up in finite time and below which all solutions are global in time as discussed by the authors.
Abstract: It is known that, for the parabolic-elliptic Keller-Segel system with critical porous-medium diffusion in dimension $\RR^d$, $d \ge 3$ (also referred to as the quasilinear Smoluchowski-Poisson equation), there is a critical value of the chemotactic sensitivity (measuring in some sense the strength of the drift term) above which there are solutions blowing up in finite time and below which all solutions are global in time. This global existence result is shown to remain true for the parabolic-parabolic Keller-Segel system with critical porous-medium type diffusion in dimension $\RR^d$, $d \ge 3$, when the chemotactic sensitivity is below the same critical value. The solution is constructed by using a minimising scheme involving the Kantorovich-Wasserstein metric for the first component and the $L^2$-norm for the second component. The cornerstone of the proof is the derivation of additional estimates which relies on a generalisation to a non-monotone functional of a method due to Matthes, McCann, \& Savare (2009).
60 citations
TL;DR: In this paper, the authors studied the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation, and they proved that there exists a t = 0 such that moments with weight exp(amin{t, 1}|v|β) are finite for t ≥ 0, where a only depends on the collision kernel and initial mass and energy.
Abstract: We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v − v *|β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v − v *|−1(v − v *)·σ and σ ∈ 𝕊 d−1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 d−1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.
TL;DR: In this article, it was shown that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation.
Abstract: We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.
TL;DR: In this paper, the Ginzburg-Landau functional is defined over a bounded and smooth three-dimensional domain and a precise asymptotic formula for the minimizing energy is derived.
Abstract: We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the magnetic field is comparable with the second critical field and that the Ginzburg-Landau parameter is large, we determine a precise asymptotic formula for the minimizing energy. In particular, this shows how bulk superconductivity decreases in average as the applied magnetic field approaches the second critical field from below. Other estimates are also obtained which allow us to obtain, in a subsequent paper [19], a fine characterization of the second critical field. The approach relies on a careful analysis of several limiting energies, which is of independent interest.
TL;DR: In this paper, the authors give a proof of asymptotic Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary pharmonic functions.
Abstract: We give a proof of asymptotic Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Lipsc...
TL;DR: In this paper, the problem of reconstructing a fully anisotropic conductivity tensor from internal functionals of the form ∇u·γ∇u where u solves ∇·(γ(u) = 0 over a given bounded domain X with pre...
Abstract: We investigate the problem of reconstructing a fully anisotropic conductivity tensor γ from internal functionals of the form ∇u·γ∇u where u solves ∇·(γ∇u) = 0 over a given bounded domain X with pre...
TL;DR: On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a constant point.
Abstract: On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a...
TL;DR: In this article, it was shown that plane wave solutions to the cubic nonlinear Schrodinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter.
Abstract: It is shown that plane wave solutions to the cubic nonlinear Schrodinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.
TL;DR: In this paper, a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ-×-ℝ d with d ≤ 6 was established.
Abstract: In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].
TL;DR: In this article, the Fourier coefficients of a solution of KdV in Sobolev space H N, N ≤ 0, admit a WKB on the circle of the circle.
Abstract: In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space H N , N ≥ 0, admit a WKB...
TL;DR: In this article, the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, dv, D 2 v, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the secondorder derivatives is established.
Abstract: We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D 2 v, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D 2 v| ≤K, where K is any given constant. For large |D 2 v| some kind of relaxed convexity assumption with respect to D 2 v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.
TL;DR: In this paper, the existence of blow-up sign-changing families on compact Riemannian manifolds has been studied, and it is shown that such families exist in two main cases: in small dimension n ǫ ∈ {3, 4, 5, 6}.
Abstract: Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (u ϵ)ϵ>0 ∈ C 2, θ(M), θ ∈ (0, 1), of solutions to where Δ g : = −div g (∇) and h ∈ C 0, θ(M) is a potential. Assuming the existence of a nondegenerate solution to the limiting equation (which is a generic assumption), we prove that such families exist in two main cases: in small dimension n ∈ {3, 4, 5, 6} for any potential h or in dimension 3 ≤ n ≤ 9 when . These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet [11] and Khuri et al. [19].
TL;DR: In this paper, the authors considered the Cahn-Hilliard equation on a manifold with conical singularities and proved the short-time solvability of the Allen-Cahn equation in Lp-Mellin-Sobolev spaces.
Abstract: We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clement and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.
TL;DR: The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations as mentioned in this paper, and an extensive study of this equation was given, proving existence, regularity, and blow-up results.
Abstract: The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.
TL;DR: In this article, the authors studied spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes, and obtained an analogue of the Povzner inequality in the special relativity case.
Abstract: In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the re...
TL;DR: On a closed manifold, this article gave a quantitative Carleman estimate on the Schrodinger operator and deduced quantitative uniqueness results for solutions to the Schodoringer equation using doubling estimates.
Abstract: On a closed manifold, we give a quantitative Carleman estimate on the Schrodinger operator. We then deduce quantitative uniqueness results for solutions to the Schrodinger equation using doubling estimates. Finally we investigate the sharpness of this results with respect to the electric potential.
TL;DR: In this article, a global bifurcation theory is used to establish a global continuum of solutions throughout which the mean-depth is a fixed quantity, and the limiting behavior of solutions in this continuum, which include the existence of weak stagnation points, is established.
Abstract: We consider steady periodic water waves with vorticity which propagate over a flat bed with a specified fixed mean-depth d > 0. Following a novel reformulation of the governing equations, we use global bifurcation theory to establish a global continuum of solutions throughout which the mean-depth is a fixed quantity. Furthermore, we establish the limiting behavior of solutions in this continuum, which include the existence of weak stagnation points, that are characteristic of large-amplitude steady periodic water waves.
TL;DR: In this article, the authors studied the initial value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables and proved that the problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data.
Abstract: We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.
TL;DR: In this article, a class of nonautonomous second order parabolic equations with unbounded coefficients defined in I × ℝ d, where I is a right-halfline, was considered.
Abstract: We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in I × ℝ d , where I is a right-halfline. We prove logarithmic Sobolev and Poincare inequal...
TL;DR: In this article, the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients was constructed, and the main goal was to construct the Green functi...
Abstract: We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. Our main goal is construct the Green functi...
TL;DR: In this article, a smooth simple curve in ℝ N, N ≤ 2, possibly with boundary is defined, and the neighborhood of the expanded curve is defined as an open normal tubular neighborhood.
Abstract: Let Γ denote a smooth simple curve in ℝ N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ∖∂Γ}. Consider the supe...