On the Construction of Cosine Operator Functions and Semigroups on Function Spaces with Generator a(x)(d2dx2)+b(x)(d/dx)+c(x); Theory

In this paper we develop a method to solve exactly partial differential equations of the type ( ⁿ /t ⁿ )f(x,t)=(a(x)( ⁿ /x ⁿ )+b(x) (/x+c(x))f(x,t); n=1,2, with several boundary conditions, where f·,t) lies in a function space. The most powerful tool here is the theory of cosine operator functions and their connection to (holomorphic) semigroups. The method is that generally we are able to unify and generalize many theorems concerning problems in the theories of holomorphic semigroups, cosine operator functions, and approximation theory, especially these dealing with approximation by projections. These applications will be found in [14].     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

A Method of Solution for the One-Dimensional Heat Equation Subject to Nonlocal Conditions

The problem of solving the one-dimensional heat equation /t - ²/x² = f(x, t) subject to given initial and nonlocal conditions is considered. It is solved in the Laplace transform domain by taking the Laplace transform of the unknown function with respect to time t. The physical solution is recovered with the help of a numerical technique for inverting the Laplace transform.AMS Subject Classification (1991): 35K20.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

Smoothability of the Conformal Boundary of a Lorentz Surface Implies 'Global Smoothability'

In 1985, Kulkarni defined the conformal boundary of a simply connected and time-oriented Lorentzian surface . He also introduced a notion of 'smoothability' of this boundary, depending only on local properties of . In this paper we show that smoothability of is in fact a global property of . In doing so, we classify Lorentzian surfaces with smoothable boundaries up to conformal homeomorphism. To be specific, suppose that the Minkowski plane E ² ₁ is the x,y-plane with metric dxdy. Our main theorem states that if is smoothable then is conformally homeomorphic to the interior U of a Jordan curve in E ² ₁ that is locally the graph of a continuous function over either the x-axis or the y-axis at each point of U.     

Parametrices for a class of differential operators with multiple characteristics

Summary In this paper we give the construction of a parametrix for a class of differential operators of the type ²/t²–t^2k+1 + t^q(/x_i), k N, q N, qk.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen

Summary For solving the nonlinear systemG(x, t)=0,G| ⁿ × ^{1 n} , which is assumed to have a smooth curve of solutions a continuation method with self-choosing stepsize is proposed. It is based on a PC-principle using an Euler-Cauchy-predictor and Newton's iteration as corrector. Under the assumption thatG is sufficiently smooth and the total derivative (₁ G(x, t)₂ G(x, t)) has full rankn along the method is proven to terminate with a solution (x ^N , 1) of the system fort=1. It works succesfully, too, if the Jacobians ₁ G(x, t) become singular at some points of , e.g., if has turning points. The method is especially able to give a point-wise approximation of the curve implicitly defined as solution of the system mentioned above.

     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

Nonlinear Diffusion with Absorption

We consider the nonlinear diffusion-absorption problem in L¹() u - div a(, Du) + j (, u) f on , u = 0 on where is an arbitrary open set in ^N. Conditions for the existence of a strong solution are presented and the existence of a natural generalized solution in the general case is established. We study the dependence of the generalized solution on the absorption term j and characterize generalized solutions which are essentially bounded.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

Complexity of the redholm equation of the second kind with kernels from classes with dominating mixed derivative

One finds the exact order of complexity of approximate solutions of Fredholm integral equations of the second kind with periodic kernels H(t,) and free terms f (t), having continuous derivatives [(^i+jH)/(t^{i j})], f⁽ⁱ⁾(t), i, .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1138–1145, August, 1990.     

The asymptotic interrelation of the Cauchy problem solutions corresponding to parabolic and telegraph type equations

Mass and heat transport processes modelled by parabolic and telegraph type equations are discussed. In order to do this the fundamental solution of the Cauchy ProblemE(x, t) for the telegraph equation (²²/t ² + 2m /t–c ²)E(x, t)=0 (xR ⁿ ,m andc are positive constants, is assumed to be a small one, the boundaries are absent) is considered. It is shown that its support may be subdivided into 4 subrogions according to the type of the asymptotic expansion. Within two of them the asymptotics ofE(x, t) is equivalent to the Poisson kernel. It is shown that the telegraph equation may be used to solve the above mentioned problems if and only ifn=1 together with the conditionsu(x, 0) 0 and u(x, 0)/t=0 imposed on the initial values. Various types of solutions corresponding to the initial data of this kind are considered and sufficient conditions for the asymptotic transition to the traditional formalism based on parabolic equations are presented. Analogous results for the asymptotic expansion of the mass flow density are also given. It is shown that the presented methods are suitable to obtain an asymptotic expansion of the solution of the Cauchy problem if the initial data functions belong toL ₁(–, ) and their supports are compact. The connection of the considered methods with those of the probability theory is outlined as well.     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Estimate of the modulus of continuity of the generalized solutions of certain singular parabolic equations

One considers singular parabolic equations of the form (u)/t–u0,where sign u is a multivalued function, equal to -I for u<0, to 1 for u>0, and to the segment [-I,I] for u=0. Such a class of equations contains, in particular, the model for the two-phase Stefan problem, the porous medium equation, and the plasma equation. For the bounded generalized solutions u(x,t) of the indicated equations (without the assumption u/L²one has established a qualified local estimate of the modulus of continuity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ins'tituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 49–71, 1985.     

Lower Subdifferentiability and Integration

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that Tf. If is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where f is quasiconvex and is the lower subdifferential ^<. This leads to the introduction of a property that is reminiscent to cyclic monotonicity. We also consider the question of the density of the domains of subdifferential operators.