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Jon Johnsen

Publications and source records attributed to Jon Johnsen.

At least 19 recordsLinked to original sources
arXiv2020

Isomorphic well-posedness of the final value problem for the heat equation with the homogeneous Neumann condition

This paper concerns the final value problem for the heat equation under the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the problem was proven well-posed in the original sense of Hadamard under an additional assumption of Hölder continuity of the source term. The point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax--Milgram operators, yielding isomorphic well-posedness for such problems. Hereby the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, resulting in a non-local compatibility condition on the data. As a novelty, a stronger version of the compatibility condition is introduced with the purpose of characterising the data that yield solutions having the regularity property of being square integrable in the generator's graph norm (instead of its form domain norm). This result allows a direct application to the considered inverse Neumann heat problem.

math.AP↗
arXiv2020

On a criterion for log-convex decay in non-selfadjoint dynamics

The short-time and global behaviour are studied for autonomous linear evolution equations defined by generators of uniformly bounded holomorphic semigroups in a Hilbert space. A general criterion for log-convexity in time of the norm of the solution is treated. Strict decrease and differentiability at the initial time results, with a derivative controlled by the lower bound of the negative generator, which is proved strictly accretive with equal numerical and spectral abscissas.

math.AP↗
arXiv2020

A class of well-posed parabolic final value problems

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.

math.AP↗
arXiv2019

Well-Posed Final Value Problems and Duhamel's Formula for Coercive Lax--Milgram Operators

This paper treats parabolic final value problems generated by coercive Lax--Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax--Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.

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arXiv2018

Characterisation of Log-Convex Decay in Non-Selfadjoint Dynamics

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

math.AP↗
arXiv2018

Final value problems for parabolic differential equations and their well-posedness

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax--Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

math.AP↗
arXiv2018

On parabolic final value problems and well-posedness

We prove that a large class of parabolic final value problems is well posed.This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is evolution equations for Lax--Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.

math.AP↗
arXiv2017

Pointwise multiplication of Besov and Triebel--Lizorkin spaces

It is shown that para-multiplication applies to a certain product $π(u,v)$ defined for appropriate temperate distributions $u$ and $v$. Boundedness of $π(\cdot,\cdot)$ is investigated for the anisotropic Besov and Triebel--Lizorkin spaces, more precisely for $B^{M,s}_{p,q}$ and $F^{M,s}_{p,q}$ with $s\in\mathbb{R}$ and $p$ and $q\in\,]0,\infty]$, though $p<\infty$ in the $F$-case. Both generic as well as various borderline cases are treated. The spaces $B^{M,s_0}_{p_0,q_0}\oplus B^{M,s_1}_{p_1,q_1}$ and $F^{M,s_0}_{p_0,q_0}\oplus F^{M,s_1}_{p_1,q_1}$ to which $π(\cdot,\cdot)$ applies are determined in the case $\max(s_0,s_1)>0$. For generic isotropic spaces $F^{s_0}_{p_0,q_0}\oplus F^{s_1}_{p_1,q_1}$ the receiving $F^{s}_{p,q}$ spaces are characterised. It is proved that $π(f,g)=f\cdot g$ holds for functions $f$ and $g$ when $f\cdot g$ is locally integrable, roughly speaking. In addition, $π(f,u)=fu$ when $f$ is of polynomial growth and $u$ is temperate. Moreover, for an arbitrary open set $Ω$ in Euclidean space, a product $π_Ω(\cdot,\cdot)$ is defined by lifting to $\mathbb{R}^n$. Boundedness of $π$ on $\mathbb{R}^n$ is shown to carry over to $π_Ω$ in general.

math.AP↗
arXiv2017

Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel--Lizorkin spaces

The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1<p<\infty$. The symbol classes treated are the uniformly estimated ones. Some precisions are given for the general definitions of trace and singular Green operators of class 0.

math.AP↗
arXiv2017

Semi-linear boundary problems of composition type in $L_p$-related spaces

The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained from the Leray--Schauder theorem or under a Landesman--Lazer condition on the data. Existence is carried over to a wide range of $L_p$-Sobolev spaces, using a non-trivial procedure to obtain a general regularity result. In fact the results are obtained in the general scales of Besov and Triebel--Lizorkin spaces.

math.AP↗
arXiv2017

On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality

A study is made of an integral identity of Helffer and Sj{ö}strand, which for some class of probability measures yields a formula for the covariance of two functions (of a stochastic variable). In comparison with the Brascamp--Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. For a certain class of measures occurring in statistical mechanics, explicit criteria for the means are found from the Persson--Agmon formula, from compactness of embeddings and from the Weyl calculus of pseudo-differential operators, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.

math-ph↗
arXiv2017

Traces of Besov spaces revisited

For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s=\frac{n}{p}-(n-1)$ and $0<p<1$ is analysed and a new dependence on the sum-exponent $q$ is found. Through examples the restriction operator defined for $s$ down to $1/p$, and valued in $L_p$, is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classical borderline $s=\frac1p$ ($1\le p<\infty$) is given a simpler proof for all $q\in\,]0,1]$, using only basic functional analysis. The new borderline results are based on corresponding convergence criteria for series with spectral conditions.

math.AP↗
arXiv2017

Traces of anisotropic Besov--Lizorkin--Triebel spaces---a complete treatment of the borderline cases

Including the previously untreated borderline cases, the trace spaces in the distributional sense of the Besov--Lizorkin--Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the trace are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the previously untreated cases. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J.~Franke and B.~Jawerth to the anisotropic scales.

math.AP↗
arXiv2017

On the Theory of Type 1,1-Operators

This dissertation concerns the pseudo-differential operators of type 1,1. These have been known especially since around 1980, when it was shown that they play an important role in the treatment of fully non-linear partial differential equations. First an account of the historical development in the area is given, including fundamental contributions due to G. Bourdaud and L. Hörmander in 1988-89, with concise remarks on the authors contributions. Secondly a detailed exposition is given of the systematic theory of type 1,1-operators, based on the general definition of such operators proposed by the author in 2008. This includes an account of how the previous extensions are generalised hereby. Moreover, the conjecture from 1978 by C. Parenti and L. Rodino that type 1,1-operators are pseudo-local is proved in this framework. It is also analysed how such operators can change the support and the spectrum of the functions they act on. Furthermore, the question of which temperate distributions such operators have in their domains have been extensively treated, and departing from Hörmander's analysis of the operators fulfilling the twisted diagonal condition or, more generally, belong to the self-adjoint subclass, such operators are shown to be everywhere defined. Finally the paradifferential splittings and the resulting boundedness in Lp Sobolev spaces, and the more general Besov and Lizorkin--Triebel spaces, are amply discussed.

math.AP↗
arXiv2017

Regularity results and parametrices of semi-linear boundary problems of product type

This short note describes the benefit one obtains from a specific construction of a family of parametrices for a class of elliptic boundary value problems perturbed by non-linear terms of product type. The construction is based on the Boutet de Monvel calculus of pseudo-differential boundary operators for the linear elliptic parts, and on paradifferential operators for the product terms.

math.AP↗
arXiv2017

Domains of type 1,1 operators: a case for Triebel--Lizorkin spaces

Pseudo-differential operators of type 1,1 are proved continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$ for $1\le p<\infty$, when of order d, and this is the largest possible domain among the Besov and Triebel--Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators.

math.AP↗
arXiv2017

Domains of pseudo-differential operators: a case for the Triebel--Lizorkin spaces

The main result is that every pseudo-differential operator of type 1,1 and order $d$ is continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$, $1\le p<\infty$, and that this is optimal within the Besov and Triebel--Lizorkin scales.The proof also leads to the known continuity for $s>d$, while for all real $s$ the sufficiency of Hörmander's condition on the twisted diagonal is carried over to the Besov and Triebel--Lizorkin framework. To obtain this, type 1,1-operators are extended to distributions with compact spectrum, and Fourier transformed operators of this type are on such distributions proved to satisfy a support rule, generalising the rule for convolutions. Thereby the use of reduced symbols, as introduced by Coifman and Meyer, is replaced by direct application of the paradifferential methods. A few flaws in the literature have been detected and corrected.

math.AP↗