Observability inequalities and measurable sets

Jone Apraiz; Luis Escauriaza; Gengsheng Wang; C. Zhang

Displaying similar documents to “Observability inequalities and measurable sets”

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...

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

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...

Spectral radius of weighted composition operators in $L^{p}$ -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator $a T_{α}$ , acting in the space $L^{p} (X,, μ)$ , the following variational principle holds: $l n r (a T_{α}) = m a x_{ν \in M ¹_{α, e}} \int_{X} l n | a | d ν$ , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M ¹_{α, e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and $_{\infty}$ -measurable function, where $_{\infty} = ⋂_{n = 0}^{\infty} α^{- n} ()$ . This considerably extends the range of validity of the above formula, which was previously known...

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...

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_{t} (t) + A u (t) = f (t, u (t))$ , $0 \leq t < τ$ with $u (τ) = φ$ , where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $φ$ and $f$ , that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$ , which is also ill-posed. Spectral truncation...

Sufficient conditions on the existence of factors in graphs involving minimum degree

Huicai Jia, Jing Lou (2024)

Czechoslovak Mathematical Journal

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For a set ${A, B, C, ...}$ of graphs, an ${A, B, C, ...}$ -factor of a graph $G$ is a spanning subgraph $F$ of $G$ , where each component of $F$ is contained in ${A, B, C, ...}$ . It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the $Q$ -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the $Q$ -spectral radius...

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

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In this work we study the problem $u_{t} {- div (| x |}^{- 2 γ} \nabla u) = λ \frac{u^{α}}{{| x |}^{2 (γ + 1)}} + f$ in $Ω \times (0, T)$ , $u \geq 0$ in $Ω \times (0, T)$ , $u = 0$ on $\partial Ω \times (0, T)$ , $u (x, 0) = u_{0} (x)$ in $Ω$ , $Ω \subset ℝ^{N}$ $(N \geq 2)$ is a bounded regular domain such that $0 \in Ω$ , $λ > 0$ , $α > 0$ , $- \infty < γ < (N - 2) / 2$ , $f$ and $u_{0}$ are positive functions such that $f \in L^{1} (Ω \times (0, T))$ and $u_{0} \in L^{1} (Ω)$ . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $α = 1$ , $1 + γ > 0$ ; (ii) the nonexistence of solutions if $α > 1$ , $1 + γ > 0$ ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik (2003)

Colloquium Mathematicae

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It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

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Let $H \subseteq Z \subseteq 2^{ω}$ . For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{ω}$ to Z give as preimages of H all Σₙ¹ subsets of $2^{ω}$ , then so do continuous injections.

Spectral radius of operators associated with dynamical systems in the spaces C(X)

Krzysztof Zajkowski (2005)

Banach Center Publications

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We consider operators acting in the space C(X) (X is a compact topological space) of the form $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = {(φ_{k})}_{k = 1}^{N}$ . We prove that $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , where Mes is the set of all probability vectors of measures $ν = {(ν_{k})}_{k = 1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into $ℝ^{d}$ . The paper deals with Y-weak cluster points ϕ̅ of the sequence $ϕ (\cdot, z_{j} (\cdot))$ in X, where $z_{j} : Ω \to ℝ^{m}$ is measurable for j ∈ ℕ and $ϕ : Ω \times ℝ^{m} \to ℝ^{d}$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{ϕ}$ , the integral $I (ϕ, ν_{x}) : = \int_{ℝ^{m}} ϕ (x, λ) d ν_{x} (λ)$ exists for $x \in Ω ∖ A_{ϕ}$ and $ϕ ̅ (x) = I (ϕ, ν_{x})$ on $Ω ∖ A_{ϕ}$ , where $ν = {ν_{x}}_{x \in Ω}$ is a measurable-dependent family of Radon probability measures on $ℝ^{m}$ .

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 $m > p > 1$ .

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $η$ supported on a compact subset of $ℝ$ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator $\nabla_{η}$ and a second order differential operator $Δ_{η}$ , with respect to $η$ . We generalize this approach to measures of the form $η : = ν + δ$ , where $ν$ is non-atomic and $δ$ is finitely supported. We determine...