A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that dH=0, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function H and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.
On a bounded domain $Ω$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.
The time dependent Eikonal equation is a Hamilton-Jacobi equation with Hamiltonian $H(P)=|P|$, which is not strictly convex nor smooth. The regularizing effect of Hamiltonian for the Eikonal equation is much weaker than that of strictly convex Hamiltonians, therefore leading to new phenomena such as the appearance of "contact discontinuity". In this paper, we study the set of singularity points of solutions in the upper half space for $C^1$ or $C^2$ initial data, with emphasis on the countability of connected components of the set. The regularity of solutions in the complement of the set of si…
We show that for any regular bounded domain $Ω\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial Ω$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.
In this short paper, we derive an alternative proof for some known [van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation*} \int_{S} \exp(tΔ)\left( f \mathbb 1_S\right)\, \mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}π} \int_{\partial S} f \,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation*} and (partially new) explicit expressions for similar expansions involving other powers of $\sqrt t$. By the same method, we also obtain short-time asymptotic…
This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators …
The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to study this problem analytically using the inverse-scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data i…
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for PDEs. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, the Mean Curvature equation, and the Eikonal operator on the graph.
A normal form for edge metrics is derived under the necessary conditions that the metric be normalized and exact. The normal forms for such an edge metric are shown to be in 1-1 correspondence with representative metrics for a reduced conformal infinity on the boundary. The normal form is constructed via solution of a singular eikonal equation at infinity. The eikonal equation is solved by proving existence and uniqueness of smooth solutions of a class of characteristic nonlinear first-order initial value problems. This is carried out by constructing a characteristic version of a Hamiltonian f…
We study the partial regularity of minimizers for certain singular functionals in the calculus of variations, motivated by Ball and Majumdar's recent modification [BM] of the Landau-de Gennes energy functional.
In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.
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.
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.
We study solutions of the first order partial differential inclusions of the form $\nabla u\in K$, where $u:Ω\subset\mathbb{R}^n\to\mathbb{R}^m$ and $K$ is a set of $m\times n$ real matrices, and derive a companion version to the result of {Müller and Šverák} [20], concerning a general linear constraint on the components of $\nabla u$. We then consider two applications: the vectorial eikonal equation and a $T_4$-configuration both under linear constraints.
For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $ω(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}ω(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamenta…
In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in \CC^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.
We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrodinger equation, the biharmonic nonlinear Schrodinger equation, the nonlinear heat equation and the nonlinear biharmonic heat equation.
We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique determination of some singular time-dependent coefficients from observations on $\partial Q$. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first cl…
The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure $μ$ as its nonhomogenous term which is given as \begin{eqnarray} -Δu&=& f(x)h(u)+μ~\text{in}~Ω,\nonumber u&=&0~\text{on}~\partialΩ,\nonumber u&>& 0~\text{on}~Ω\nonumber, \end{eqnarray} where $Ω$ is a bounded domain of $\mathbb{R}^N$, $f$ is a nonnegative function over $Ω$.
We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.
In this note we show that the general theory of vector valued singular integral operators of Calderón-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliers and dyadic homogeneous singular integral operators.
In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal condition. Singularity means that at the final time, the value of the solution can be equal to infinity. Different types of regularity of this viscosity solution are investigated: Sobolev, H{ö}lder or strong regularity.
We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.
We consider the Wulff-type energy functional $$ \mathcal{W}_Ω(u) := \int_ΩB(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.
Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }Ω. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $Ω$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-δ}+u^{-γ}),$ $δ,γ>0$ with $Ω=\mathbb{R}^N$ respectively.
This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.
We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.
We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.
We consider equation $-Δu+f(x,u)=0$ in smooth bounded domain $Ω\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $Ω\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partialΩ$. We find the condition on the order of degeneracy of $f(x,r)$ near $\partialΩ$, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $\partialΩ$. Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.
We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image $ϕ(\partialΩ)$ of a reference set $\partialΩ$ and we present some real analyticity results for the dependence upon the map $ϕ$. Then we introduce a perforated domain $Ω(ε)$ with a small hole of size $ε$ and we compute power series expansions that describe the layer potentials on $\partialΩ(ε)$ when the parameter $ε$ approximates the degenerate value $ε=0$.
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +$\infty$ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.
The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(…
The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to SPDE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamical systems it is proven that the attractor consists of a single random point. In case of real, linear multiplicative noise finite time extinction is obtained. Applications include stochastic generalized fast diffusion equations and stocha…
We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =Δu^m - \nabla(\frac{u}{v}\nabla v),&{} x\inΩ,\ t>0, \medskip v_t =Δv -uv,&{}x\inΩ,\ t>0, \medskip \frac{\partial u}{\partial ν}=\frac{\partial v}{\partialν}=0,&{}x\in\partialΩ,\ t>0, \medskip u(x,0)=u_0(x),\ v(x,0)=v_0(x), &{}x\inΩ, \end{array} \end{cases} \end{eqnarray*} in a smooth bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$. In this work it is shown that for all reasonably regular initial data $u_0\geq0$ and $v_0>0$, the corresponding Neumann initial-boundary value problem possesses a glo…
In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 Δu + u = u^p, &\text{in}Ωu > 0, &\text{in}Ω\frac{\partial u}{\partial ν} = 0,& \text{on}\partial Ω{cases} {align*} for $\varepsilon$ small and $p$ subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.
The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to \begin{equation*} Δ_{\mathbb{H}^n}^2 u=\frac{f(ξ,u)}{ρ(ξ)^a}\,\,\text{ in }Ω,\,\, u|_{\partialΩ}=0=\left.\frac{\partial u}{\partial ν}\right|_{\partialΩ}, \end{equation*} where $0\in Ω\subseteq \mathbb{H}^4$ is a smooth bounded domain, $0\leq a<Q,\,(Q=10).$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.
We study a family of singularly perturbed $q-$difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter $ε$. Moreover, we achieve the existence of a common formal power series in $ε$ which represents each actual solution, and establish $q-$Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya Theorem regarding $q-$Gevrey asymptotics. A particular Dirichlet like series is studied on the way.
We investigate the existence of local holomorphic solutions $Y$ of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety $Θ$ in $\mathbb{C}^{2}$. The coefficients are written as linear combinations of powers of a solution $X$ of some first order nonlinear partial differential equation following an idea we have initiated in a previous work \cite{mast}. The solutions $Y$ are shown to develop singularities along $Θ$ with estimates of exponential type depending on the growth's rate of $X$ near the singular variety. We construct t…
Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we pre- dict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two …
We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter $\varepsilon$. For the analysis, we consider the system equations as partial differential-algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order es…
We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular $p$-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient field…
