Inequalities involving heat potentials and Green functions

Neil A. Watson

Displaying similar documents to “Inequalities involving heat potentials and Green functions”

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D \subset ℝ^{d}$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$ . We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$ . Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$ , we provide results concerning the well-posedness...

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{- β}$ , with $β \in [0, \infty)$ . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$ . If $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

Classical boundary value problems for integrable temperatures in a $C^{1}$ domain

Anna Grimaldi Piro, Francesco Ragnedda (1991)

Annales Polonici Mathematici

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Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^{1}$ -base and data in $h_{c}^{1}$ , a subspace of $L$ 1. We derive our results, considering the action of an adjoint operator on $B_{T} M O C$ , a predual of $h_{c}^{1}$ , and using known properties of this last space.

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

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Let $(X, d, μ)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $μ$ . Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^{2} (X)$ . Assume that the semigroup $e^{- t L}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_{L}^{p} (X)$ be the Hardy space associated with $L .$ We show the boundedness of Stein’s square function $𝒢_{δ} (L)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_{L}^{p} (X)$ to $L^{p} (X)$ , and also study...

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

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We consider solutions of quasilinear equations $u_{t} = Δ u^{m} + u^{p}$ in $ℝ^{N}$ with the initial data $u_{0}$ satisfying $0 < u_{0} < M$ and ${lim}_{| x | \to \infty} u_{0} (x) = M$ for some constant $M > 0$ . It is known that if $0 < m < p$ with $p > 1$ , the blow-up set is empty. We find solutions $u$ that blow up throughout $ℝ^{N}$ when $m > p > 1$ .

On the potential theory of some systems of coupled PDEs

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri, Sabah Haddad (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1} u = - μ_{1} v$ , $L_{2} v = - μ_{2} u$ , on a domain $D$ of $ℝ^{d}$ , where $μ_{1}$ and $μ_{2}$ are suitable measures on $D$ , and $L_{1}$ , $L_{2}$ are two second order linear differential elliptic operators on $D$ with coefficients of class $𝒞^{\infty}$ . We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_{1}$ and $L_{2}$ , and...

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials $V_{μ}$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{μ}$ are defined by $V_{μ} (z) = \int_{B} l o g (1 / | ϕ_{z} (w) |) d μ (w)$ , where for a fixed z ∈ B, $ϕ_{z}$ denotes the holomorphic automorphism of B satisfying $ϕ_{z} (0) = z$ , $ϕ_{z} (z) = 0$ and $(ϕ_{z} \circ ϕ_{z}) (w) = w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

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For $1 \leq n < d$ integers and $ρ > 2$ , we prove that an $n$ -dimensional Ahlfors-David regular measure $μ$ in $ℝ^{d}$ is uniformly $n$ -rectifiable if and only if the $ρ$ -variation for the Riesz transform with respect to $μ$ is a bounded operator in $L^{2} (μ)$ . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^{2} (μ)$ boundedness of the Riesz transform to the uniform rectifiability of $μ$ .

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...