This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.
We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.
We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.
Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -Δ_{p}u+|\nabla u|^q=0, \ \hbox{in} \ (0,\infty)\times\real^N, {equation*} where $2N/(N+1)
We are concerned with the mean field equation with singular data on bounded domains. Under suitable non-degeneracy conditions we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at singular points. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and suitably defined Pohozaev-type identities.
In this paper, we consider a nonlinear Fuchsian type partial differential equation of the second order in the complex domain. Under a very weak assumption, we show the uniqueness of the solution. The result is applied to the problem of analytic continuation of local holomorphic solutions of this equation.
The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.
Given $Ω$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-Δ\,u\,=\,\frac{f}{u^β}$ in $H^1_{loc}(Ω)$, under zero Dirichlet boundary conditions. For $β>0$ and $f\in L^1(Ω)$, we prove that the solution is unique.
In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,β}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using that regularity, a strict comparison principle and Sard's theorem.
In this article, we investigate the existence and uniqueness of weak solutions to the continuous coagulation equation with collisional breakage for a class of unbounded collision kernels and distribution function. The collision kernels and distribution functions may have a singularity on both the coordinate axes. The proof of the existence result is based on a classical weak L^1 compactness method applied to suitably chosen to approximate equations. The question of uniqueness is also shown for some restricted class of collision kernels.
In this paper we consider the inverse boundary value problem for the Schrödinger equation with potential in $L^p$ class, $p>4/3$. We show that the potential is uniquely determined by the boundary measurements.
For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4π$ we prove that bubbling solutions are unique under non-degeneracy assumptions. This work extends a previous work of Bartolucci, et, al \cite{bart-4}.
We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy-Littlewood-Sobolev type functional in…
We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $ε$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $ε$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan matrices and eigenvalue analysis of linearized Liouville-type problems. This seems to be the first result for this class of problems and it covers all the Lie algebras of any rank.
We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.
In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
The existence of a solution to the two dimensional incompressible Euler equations in singular domains was established in [Gérard-Varet and Lacave, The 2D Euler equation on singular domains, submitted]. The present work is about the uniqueness of such a solution when the domain is the exterior or the interior of a simply connected set with corners, although the velocity blows up near these corners. In the exterior of a curve with two end-points, it is showed in [Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl \textbf{26} (2009), 1121-11…
This paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.
In this paper, existence and uniqueness of solutions to a non-linear, initial value problem is studied. In particular, we consider a special type of problem which physically represents the time evolution of particle number density resulted due to the coagulation and fragmentation process. The coagulation kernel is chosen from a huge class of functions, both singular and non-singular in nature. On the other hand, fragmentation kernel includes practically relevant non-singular unbounded functions. Moreover, both the kernels satisfy a linear growth rate of particles at infinity. The existence the…
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.
In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calderón problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.
We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$ boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.
We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -Δu= h(u){f} \ \ \text{in}\,\ Ω, $$ where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.
Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, b) for initial data above this threshold function $…
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.
We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ide…
We prove global existence and uniqueness of solutions to a Cahn-Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the the diffusion operator …
In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.
In this paper we prove the existence and uniqueness of positive classical solution of the fractional Laplacian with singular nonlinearity in a smooth bounded domain with zero Drichlet boundary conditions. By the method of sub-supersolution, we derive the existence of positive classical solution to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solution for the fraction Laplacian. Thanks to \cite{XY}, the regularity follows from the boundedness of weak solution.
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 \[ \begin{cases} -Δ_{p}u-{\displaystyle \fracμ{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+λ|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<μ<((N-p)/p)^{p}$, $0\le s<p$ and $λ\in\mathbb{R}$. A related limiting problem is also considered.
In dimension $n\geq 3$, we prove a local uniqueness result for the potentials $q$ of the Schrödinger equation $-Δu+qu=0$ from partial boundary data. More precisely, we show that potentials $q_1,q_2\in L^\infty$ with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where $q_1\geq q_2$ and $q_1\not\equiv q_2$.
We consider the boundary value problem $-Δ_p u_λ-Δ_q u_λ=λg(x) u_λ^{-β}$ in $Ω$ , $u_λ=0$ on $\partial Ω$ with $u_λ>0$ in $Ω.$ We assume $Ω$ is a bounded open set in $\mathbb{R}^N$ with smooth boundary, $1<p<q<\infty$, $β\in [0,1),$ $g$ is a positive weight function and $λ$ is a positive parameter. We derive an estimate for $u_λ$ which describes its exact behavior when the parameter $λ$ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear ell…
In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when $u$ is a non-trivial solution to $\triangle u + W \cdot \nabla u + V u = 0$ in some open, connected subset of $\mathbb R^n$, where $n \geq 3$, we characterize the vanishing order of solutions in terms of the norms of $V$ and $W$ in their respective Lebesgue spaces. Using these maximal order of vanishing estimates, we also establish quantitat…
We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted $\mathbf{R}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-…
A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is proven. No special geometrical condition is imposed on the inaccessible part of the boundary of the domain, apart from imposing that the boundary of the domain is $C^{1,1}$. The coefficients are assumed to coincide on a neighbourhood of the boundary, a natural property in applications.
Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice; in the balance equations of microforces and microenergy, the two unknowns are the order parameter and the chemical potential. A simpler version of the same system has recently been discussed in arXiv:1103.4585v1. In this paper, a fairly more general phase-field equation is…
We address the question of the uniqueness of solution to the initial value problem associated to the equation \partial_{t}u+iα\partial^{2}_{x}u+β \partial^{3}_{x}u+iγ|u|^{2}u+δ|u|^{2}\partial_{x}u+εu^{2}\partial_{x}\bar{u} = 0, \quad x,t \in \R, and prove that a certain decay property of the difference $u_1-u_2$ of two solutions $u_1$ and $u_2$ at two different instants of times $t=0$ and $t=1$, is sufficient to ensure that $u_1=u_2$ for all the time.
In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely $ \displaystyle u\geq 0 \mbox{ in } Ω$, $ \displaystyle - div \,A(x) D u = F(x,u) \mbox{ in} \; Ω$, $u = 0 \mbox{ on} \; \partial Ω$, with $F(x,s)$ a Carathéodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{Γ(s)}\,\,\mbox{ a.e. } x\inΩ,\, \forall s>0, $$ with $h$ in some $L^r(Ω)$ and $Γ$ a $C^1([0,+\infty[)$ function such that $Γ(0)=0$ and $Γ'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition a…
We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-Δu+u^q=0$ ($q>1$) in $Ω\times (0,\infty)$, $u=\infty$ on $\partialΩ\times (0,\infty)$ and $u(.,0)\in L^1(Ω)$, when $Ω$ is a bounded domain in $\mathbb R^N$. We construct a maximal solution, prove that this maximal solution is a large solution whenever $q<N/(N-2)$ and it is unique if $\partialΩ=\partial\barΩ^c$. If $\partialΩ$ has the local graph property, we prove that there exists at most one solution to problem (P)
