Parabolic PDEs with Dynamic Data under a Bounded Slope Condition.

Parabolic equations and the bounded slope condition

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied 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MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.

Lipschitz minimizers for a class of integral functionals under the bounded slope condition

In this paper, we study the Cauchy–Dirichlet problem ∂tu-divDξf(t,Du)=0inΩT,u=uoon∂PΩT,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _t u - {\ ext {div}} \\left( D_\\xi f(t, Du)\\right) = 0 &{} \\quad \\hbox {in} \\ \\Omega _T, \\\\ u = u_o &{} \\quad \\hbox { on} \\ \\partial _{\\mathcal {P}} \\Omega _T,\\\\ \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega subset mathbb {R}^n is a convex and bounded domain, f:[0,T]times {mathbb {R}}^n rightarrow {mathbb {R}} is L^1-integrable in time and convex in the second variable. Assuming that the initial and boundary datum u_o:{overline{Omega }}rightarrow {mathbb {R}} satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.

Chapter 6 Application of Monotone Type Operators to Parabolic and Functional Parabolic PDE's

Existence and uniqueness of CMC parabolic graphs in [formula omitted] with boundary data satisfying the bounded slope condition

In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation in a bounded domain with boundary problem. An expansion formula for fractional derivative, optimal conditions and a new solution scheme is proposed.

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form over the functions that assume given boundary values ϕ on ∂Ω. The vector field satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math. 115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions when ϕ satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511–530] has introduced a new type of hypothesis on the boundary condition ϕ: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if ϕ is the restriction to ∂Ω of a convex function. We show that if a and F satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on Ω.

We study first-order symmetrizable hyperbolic N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\ imes N$$\\end{document} systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=0$$\\end{document}, these systems take the form ∂tu+A(t,x,y,xDx,Dy)u=f(t,x,y),(t,x,y)∈(0,T)×R+×Rd,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u + {{\\mathcal {A}}}(t,x,y,xD_x,D_y) u = f(t,x,y), \\quad (t,x,y)\\in (0,T)\ imes {{\\mathbb {R}}}_+\ imes {{\\mathbb {R}}}^d, \\end{aligned}$$\\end{document}where A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is a first-order differential operator with coefficients smooth up to x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=0$$\\end{document} and the derivative with respect to x appears in the combination xDx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$xD_x$$\\end{document}. No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator ∂t+A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t + {{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is well-posed in that scale. More specifically, solutions u exhibit formal asymptotic expansions of the form u(t,x,y)∼∑(p,k)(-1)kk!x-plogkxupk(t,y)asx→+0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u(t,x,y) \\sim \\sum _{(p,k)} \\frac{(-1)^k}{k!}x^{-p} \\log ^k \\!x \\, u_{pk}(t,y) \\quad \\hbox { as}\\ x\\rightarrow +0 \\end{aligned}$$\\end{document}where (p,k)∈C×N0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p,k)\\in {{\\mathbb {C}}}\ imes {{\\mathbb {N}}}_0$$\\end{document} and ℜp→-∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re p\\rightarrow -\\infty $$\\end{document} as |p|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|p|\\rightarrow \\infty $$\\end{document}, provided that the right-hand side f and the initial data u|t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u|_{t=0}$$\\end{document} admit asymptotic expansions as x→+0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\rightarrow +0$$\\end{document} of a similar form, with the singular exponents p and their multiplicities unchanged. In fact, the coefficients upk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{pk}$$\\end{document} are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients upk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{pk}$$\\end{document} solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator ∂t+A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t+{{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is well-posed in the scale of standard Sobolev spaces Hs((0,T)×R+1+d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s((0,T)\ imes {{\\mathbb {R}}}_+^{1+d})$$\\end{document}.

1. Introduction. During the past few years much work has been devoted to the problem of characterizin g sets which are invariant with respect to a given ordinary differential equation. More recently the papers [ 2], [ 15] have addressed themselves to the same question for nonlinear parabolic differential equations. The purpose of this paper is twofold. First we provide some exten­ sions of invariance results (for parabolic equations) (sections 3 and 4) and secondly show that the assumptions which are sufficient for a given region to be invariant also yields existence of solutions of initial boundary value problems. We further show that the systems con­ sidered have the classical Hukuhara-Kneser property, i.e., the set of solutions of a given initial boundary value problem is a continuum in an appropriate function space; we thus provide an extension of a result of [5] to a large class of systems of parabolic differential equations. Our invariance results were motivated by a result of [ 1], where certain geometric conditions were given to establish the solvability of two point boundary value problems for systems of second order ordinary differential equations; the type of result given there is the following: Given a nonempty bounded open convex set such that the vector field defined by the nonlinear terms in the differential equation never points into the interior of the convex set, then for any two points in the convex set there exists a solution of the equation connecting the two points and which has values in that set. It is precisely these conditions that were adopted in [15] to show that they implied invariance for a system of parabolic equations. Under somewhat weaker assumptions than in [ 15] we not only prove invariance of that convex region but also demonstrate existence of solutions. Using some ideas suggested by [11] we further show that essentially the same type of result holds for convex sets with empty interior. In order to establish the Hukuhara-Kneser property for systems of parabolic equations satisfying our conditions we rely on results and ideas about the structure of the set of fixed points of completely con

In this paper we introduce a purely variational approach to the gradient flows, naturally arising in image denoising models, yielding the existence of global parabolic minimizers, in the sense that $ \int_0^{T}\big[\int_\Omega u \partial_t\varphi\, dx + F(u)\big] \,dt \le \int_0^{T}F(u+\varphi) \,dt, $ whenever $T>0$ and $\varphi\in C_0^\infty(\Omega\times(0,T))$. Our method applies to a wide class of nonparametric regression models in image restoration analysis, such as quantile, robust, and logistic regression. A prototype functional $F$ is the by now classical ${\rm TV}(L^2)$-functional (i.e., the pure ${\rm TV}$-denoising case in image reconstruction) introduced by Rudin, Osher and Fatemi [Phys. D, 268 (1992), pp. 259--268]: $ \boldsymbol F(u):= \mathbf{TV}(u)+\frac{\kappa}{2}\int_\Omega |u-u_o|^2\, dx, $ where $u_o\colon\Omega\to [0,1]$ is a noisy, monochromatic image and $\kappa\gg 1$ a large penalization parameter. The evolutionary variational solutions are obtained as limits of maps, minimizing a convex variational functional in n+1 dimensions with domain $\Omega_T:=\Omega \times (0,T)$. Our approach yields a new way of proving the existence of global weak solutions to the associated Cauchy--Dirichlet problem, $\partial _{t}u- {\rm div} \big(\frac{D u}{|D u|}\big)=\kappa (u-u_o)$ in $\Omega \times (0,\infty)$ and $u=u_o$ on the parabolic boundary. Our approach also applies in situations where the considered functionals do not allow the derivation of the associated parabolic equation. We are able to deal with Dirichlet and Neumann type boundary conditions on the lateral boundary, and furthermore with the gradient flow associated to functionals modeling image deblurring, such as $ F(u)=\mathbf{TV}(u)+\frac\kappa2\int_\Omega\big| \mathbf K[u]-u_o\big|^2\, dx, $ where $\mathbf K\colon L^1(\Omega) \to L^2(\Omega)$ is a bounded, linear, injective operator satisfying the DC-condition $\mathbf K[1]=1$.

A procedure is described for reducing dynamical equations in two-space variables defined in a rectangle with nonhomogeneous time dependent boundary data to equations with homogeneous boundary data. The procedure applies not only to a single equation but to systems of equations with systems of boundary conditions. One-space dimensional problems are treated separately and a condition for applicability is developed in this case. Examples are presented in which dynamical equations in one and two-space variables with nonhomogeneous time dependent boundary data are reduced to equations with homogeneous boundary data. Specific applications to problems in one-space variable include a simple beam, a laminated composite plate, and a Timoshenko beam. For two-space variable problems defined in a rectangle, the application of the procedure is made to an antisymmetric angle-ply circular cylindrical panel.

A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.