Hypoellipticity of certain degenerate elliptic boundary value problems

In this paper we study the strong maximum principle for globally hypoelliptic differential operators of second-order, and reveal the underlying analytical mechanism of propagation of maximums in terms of the Lie algebra generated by diffusion vector fields and the Fichera function. Our formulation of the strong maximum principle is coordinate-free. The results here may be applied to questions of uniqueness for degenerate elliptic boundary value problems on a manifold. Furthermore, the mechanism of propagation of maximums plays an important role in the interpretation and study of Markov processes from the viewpoint of functional analysis.

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. Our degenerate boundary conditions include as particular cases the isothermal condition (Dirichlet condition) and the adiabatic condition (Neumann condition). We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. More precisely, we give sufficient conditions for our semilinear boundary value problems to have three positive solutions, which suggests that the bifurcation curves are S-shaped.

Previous article Next article On the Numerical Solution of Degenerate Linear and Nonlinear Elliptic Boundary Value ProblemsHarold J. KushnerHarold J. Kushnerhttps://doi.org/10.1137/0705053PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 Google Scholar[2] Ronald A. Howard, Dynamic programming and Markov processes, The Technology Press of M.I.T., Cambridge, Mass., 1960viii+136 MR0118514 0091.16001 Google Scholar[3] J. H. Eaton and , L. A. Zadeh, Optimal pursuit strategies in discrete-state probabilistic systems, Trans. ASME Ser. D. J. Basic Engrg., 84 (1962), 23–29 MR0153510 CrossrefGoogle Scholar[4] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons Inc., New York, N.Y., 1950xii+419 MR0038583 0039.13201 Google Scholar[5] H. J. Kushner, Methods for the numerical solution of degenerate linear and non-linear elliptic boundary value problems, Tech. Rep., 67-7, Center for Dynamical Systems, Brown University, Providence, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails The Numerical Solution of Singular Perturbations of Boundary Value ProblemsFred W. DorrSIAM Journal on Numerical Analysis, Vol. 7, No. 2 | 14 July 2006AbstractPDF (2164 KB)Vacancy Transients During Impurity Diffusion in SemiconductorsPhysical Review B, Vol. 1, No. 6 | 15 March 1970 Cross Ref BIBLIOGRAPHYIterative Solution of Nonlinear Equations in Several Variables | 1 Jan 1970 Cross Ref Numerical methods for the solution of the degenerate nonlinear elliptic equations arising in optimal stochastic control theoryIEEE Transactions on Automatic Control, Vol. 13, No. 4 | 1 Aug 1968 Cross Ref Volume 5, Issue 4| 1968SIAM Journal on Numerical Analysis647-889 History Submitted:15 January 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705053Article page range:pp. 664-679ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

(1991). Lp and Holder estimates for a class fo degenerate elliptic boundary value problems Application to the Monge–Amp ere equation. Communications in Partial Differential Equations: Vol. 16, No. 6-7, pp. 997-1031.

Part 1 Composite type equations as the original mathematical object: equations of the composite type in mathematics and mathematical simulation canonical types of third order boundary-value problems for third order equations of composite type hyperbolic boundary-value problem uniqueness of regular solutions existence of the regularized solutions of the hyperbolic boundary-value problem the elliptic boundary-value problem uniqueness of regular solutions existence of generalized solutions of the elliptic boundary-value problem existence of regular solutions of the elliptic boundary-value problem correlation of the hyperbolic and elliptic boundary-value problems remark on the third order equations of the variable direction nonlocal problems for equations of composite type. Part 2 Solvability of inverse problems and other applications: inverse problems for equations with partial derivatives linear inverse problems for elliptic and parabolic equations on solvability of nonlinear inverse problems solvability of nonclassical boundary-value problems for equations of second order and third order the mixed problem for second order equations not solved for the time derivative boundary-value problems for the second order equations of the mixed type the problem with oblique derivative for equations of second and third orders a problem of viscous elasticity and the third order equation in noncylindrical domains connected with this problem.

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.

In this paper we give, for the first time, an abstract interpretation of such boundary value problems for elliptic equations of the second order that a part of boundary value conditions contains also a differentiation of the second order. Boundary value problems for elliptic equations are reduced to the boundary value problem for a system of differential- operator equations (see below problem (1)− (3)). A solution of this system is not a vector- function but one function. At the same time, the system is not overdetermined. Boundary value problems for elliptic equations (see section 3), which we consider in this paper, do not satisfy the Lopatinskii condition (so, the problems are irregular) since the tangent derivative is given on the domain boundary.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. 
 Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems.
 Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. 
 Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations.
 Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

In a rectangular domain, the second boundary value problem for the Viscous Transonic Equation is considered. The uniqueness of the solution to the problem is proved using the energy integral method. The existence of a solution is proved by the method of separation of variables, i.e. it is sought in the form of a product of two functions X (x) and Y (y). For definition Y (y), an ordinary differential equation of the second order with two boundary conditions on the boundaries of segment [0,q] is obtained. For this problem, the eigenvalues and the corresponding eigenfunctions are found at n ∈ N. For definition X (x), an ordinary differential equation of the third order with three boundary conditions on the boundaries of segment [0,q] is obtained. The solution to this problem is found in the form of an infinite series, uniform convergence, and the possibility of term-by-term differentiation under certain conditions on the given functions is proven. The convergence of the second-order derivative of the solution with respect to variable y is proved using the Cauchy-Bunyakovsky and Bessel inequalities. When substantiating the uniform convergence of the solution, the absence of a “small denominator” is proved.

A minor modification of the Clifford Cauchy kernel is used to build up an operator calculus. This calculus works extremely well in investigating boundary value problems of elliptic partial differential equations using methods of Clifford analysis. As examples the Laplace equation and the Stokes’ system are considered here.

Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case do not extend to the hyperbolic case. Extensions to singular boundaries and mixed problems are included. The dichotomy referred to in the title pertains to the different but parallel Sturmian theories for elliptic and hyperbolic boundary problems: Results of similar type are true for both types of problems and yet other results true in the elliptic case are false in the hyperbolic case.

The mixed problem for a nonlinear degenerate elliptic equation of second order

In this work, we study the regularization method for solving the Boundary Value Problem (BVP) for heat equation. The discretization method applied with two variables on Volterra integral equation in order to covert the problem into a linear operator equation after applied the separation of variables method to solve the partial differential equation. The regularization way used to obtain the estimate solution by using the Lavrentiev regularization method.

We analyze different statements of the boundary value problem for the convection equations in order to use Ostroumov–Birikh type exact solutions for describing the evaporative convection regimes in two-layer systems of liquids and gas–vapor mixtures filling a plane channel. The statements differ in the types of boundary conditions for the vapor concentration and temperature functions on the channel walls. We discuss the boundary value problems modeling two-layer flows in the channel, describe the characteristics of hydrodynamic, thermal, and concentration fields for the working HFE7100-nitrogen system and classify the types of flows by distinguishing three classes: purely thermocapillary, mixed, and Poiseuille type flows.

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.