There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case.
There is considered the category of second order parabolic equations posed on arbitrary manifolds. In this category, for the given nonlinear heat equation we could find morphisms from it to other parabolic equations with the same or a smaller number of independent variables. This allows to receive some classes of solutions of original equation from the class of all solutions of such a reduced equation. Classification of morphisms (with the selection from every equivalence class of the simplest "canonical" representatives) is carried out. There are derived the necessary and sufficient conditions for canonical morphisms of heat equation to the parabolic equation on the other manifold. These conditions are formulated in the differential geometry language.
The comparison with invariant solutions classes, obtained by the Lie group methods, is carried out. It is proved that discovered solution classes are richer than invariant solution classes, even if we find any (including discontinuous) symmetry groups of original equation.
We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.
In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.
From the uniformization theorem, we know that every Riemann surface has a simply-connected covering space. Moreover, there are only three simply-connected Riemann surfaces: the sphere, the Euclidean plane, and the hyperbolic plane. In this paper, we collect the known heat kernels, or Green's functions, for these three surfaces, and we derive the differential forms heat kernels as well. Then we give a method for using the heat kernels on the universal cover to generate the heat kernel on the underlying surface.
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are $A_3$ and $A_5$ (in the classification of Arnol'd's school). We show that in the non-gene…
We study the solvability of the initial value problem for the semilinear heat equation $u_t-Δu=u^p$ in a Riemannian manifold $M$ with a nonnegative Radon measure $μ$ on $M$ as initial data. We give sharp conditions on the local-in-time solvability of the problem for complete and connected $M$ with positive injectivity radius and bounded sectional curvature.
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy problem for the inhomogeneous heat equation, when the free term lies in a weighted parabolic Hölder or Sobolev space with discrete asymptotics. This generalizes a result previously obtained by Coriasco, Schrohe, and Seiler (Thm. 7.2 in Math. Z. 244 (2003), 235--269) by different mean…
We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to the previously known results in negatively curved contexts, the radiality assumption on the initial data is not required. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. Moreover, we provide a counterexample such that this asymptotic phenomenon fails in sup norm on man…
We show that any closed n-dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a lower bound on the Ricci curvature, the injectivity radius and the volume. It follows that the manifold can be embedded by a finite number of eigenfunctions of the Laplace operator. Again, this number only depends on the geometric bounds and the dimension. In addition, both maps can be made arbitrarily close to an isometry. In the appendix, we derive quantitat…
Bounds on the logarithmic derivatives of the heat kernel on a compact Riemannian manifolds have been long known, and were recently extended, for the log-gradient and log-Hessian, to general complete Riemannian manifolds. Here, we further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition on the distance to infinity available, for derivatives of all orders. Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we also allow the addition of a conservative vector field. We show that these bounds are sharp in …
We investigate heat kernel estimates of the form $p_{t}(x, x)\geq c_{x}t^{-α},$ for large enough $t,$ where $α$ and $c_{x}$ are positive reals and $c_{x}$ may depend on $x,$ on manifolds having at least one end.
We study a multi-marginal optimal transportation problem on a Riemannian manifold, with cost function given by the average distance squared from multiple points to their barycenter. Under a standard regularity condition on the first marginal, we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable, thus inducing a Monge solution. This result generalizes McCann's polar factorization theorem on manifolds from 2 to several marginals, in the same sense that a well known result of Gangbo and Swiech generalizes Brenier's polar factorization the…
We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville--type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.
Local well-posedness for the nonlinear heat and Schrödinger equations with nonlinear Wentzell boundary conditions on time-dependent compact Riemannian manifolds
We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.
We prove Fatou type theorems for solutions of the heat equation in sub- Riemannian spaces. The doubling property of L-caloric measure, the Dahlberg estimate, the local comparison theorem, among other results, are established here. A backward Harnack inequality is proved for non-negative solutions vanishing in the lateral boundary.
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
Estimates are obtained for the expected volume of intersection of independent Wiener sausages in Euclidean space in the small time limit. The asymptotic behaviour of the weighted diagonal heat kernel norm on compact Riemannian manifolds with smooth boundary is obtained in the small time limit
In this paper, we develop a factorization method to reconstruct cavities in a heat conductor by knowing the Neumann-to-Dirichlet map at the boundary of this conductor. The factorization method is a very well known reconstruction method for inverse problems described by self-adjoint elliptic equations. It enables us to reconstruct the boundaries of unknown targets such as cavities and inclusions. This method was successfully applied for examples to the inverse scattering problem by obstacles for the acoustic wave and the electrical impedance tomography problem. The forward model equations for t…
The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.
Let $Ω$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $Ω$ has infinite measure, and if $Ω$ has finite heat content $H_Ω(T)$ for some $T>0$, then $H_Ω(t)<\infty$ for all $t>0$. Comparable two-sided bounds for $H_Ω(t)$ are obtained for such $Ω$.
A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a Riemann-Hilbert problem using the Hurd's method. Secondly, the Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient o…
By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $α_{1}$ and $α_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($α_{1}=…
Let $Ω$ be an open set in a geodesically complete, non-compact, $m$-dimen-sional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary. We study the heat flow from $Ω$ into $M-Ω$ if the initial temperature distribution is the characteristic function of $Ω$. We obtain a necessary and sufficient condition which ensures that an open set $Ω$ with infinite measure has finite heat content for all $t>0$. We also obtain upper and lower bounds for the heat content of $Ω$ in $M$. Two-sided bounds are obtained for the heat loss of $Ω$ in $M$ if the measure of $Ω$ is finite.
This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main result proves that the total variation of a function equals the limit characterizing the space BV. The proof is carried out following two fully independent approaches, a probabilistic and an analytic one. Each method presents different advantages.
Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the initial temperature distribution, and the specific heat of D are given by $r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary, and 1<a<2, 1<b<2.
We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series: \begin{equation} Q_Ω(t) = \sum_{k=0}^{\infty} a_k t^{k/2}, \qquad \text{as } t\to 0. \end{equation} We compute explicitly the coefficients up to order $k=5$, in terms of sub-Riemannian invariants of the domain and its boundary. Furthermore, we prove that every coefficient can be obtained a…
We study the heat equation $\frac{\partial u}{\partial t}-Δu=0,\ u(x,0)=ω(x),$ where $Δ:=dd^{*}+d^{*}d$ is the Hodge laplacian and $u(\cdot ,t)$ and $ω$ are $p$-differential forms in the complete Riemannian manifold $(M,g).$ Under weak bounded geometrical assumptions we get estimates on its semigroup of the form: acting on $p$-forms with $p\geq 1$ and $k\geq 0$: $\displaystyle \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-tΔ_{p}}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k).$ Acting on functions, i.e. with $p=0,$ we get a better result: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left…
In a cylinder $D_T = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$, we examine the relation between the $L$-caloric measure, $dω^{(x,t)}$, where $L$ is the heat operator associated with a system of vector fields of Hörmander type, and the measure $dσ_X\times dt$, where $dσ_X$ is the intrinsic $X$-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain $Ω$ we establish the mutual absolute continuity of $dω^{(x,t)}$ and $dσ_X…
The hydrodynamic escape problem (HEP), which is characterized by a free boundary value problem of Euler equation with gravity and heat, is crucial for investigating the evolution of planetary atmospheres. In this paper, the global existence of transonic solutions to the HEP is established using the generalized Glimm method. The new version of Riemann and boundary-Riemann solvers, are provided as building blocks of the generalized Glimm method by inventing the contraction matrices for the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm-Goodman wave interaction estimates …
We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a …
In this paper we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis that permits to transform the hypoelliptic heat equation in a one dimensional heat equation with a quartic potential.
For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to …
Let M be a compact Riemannian manifold and let $μ$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $μ$ to a measure $ν$: each S factors uniquely a.e. into the composition S = T \circ U, where U : M -> M is volume preserving and T : M -> M is the optimal map transporting $μ$ to $ν$ with respect to the cost function d^2/2. In this article we study the polar factorization of conformal and projective maps of the sphere S^n. For conformal maps, which m…
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.
We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric.…
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented r…
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncom…
This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic …
