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Papers by G.n Hile

Research paper thumbnail of Liouville theorems for even order elliptic systems

Indiana University Mathematics Journal

Research paper thumbnail of Liouville theorems for nonlinear parabolic equations of second order

Differential and Integral Equations

We study entire solutions (i.e., solutions defined in all of R n+1) of second-order nonlinear par... more We study entire solutions (i.e., solutions defined in all of R n+1) of second-order nonlinear parabolic equations with linear principal part. Under appropriate hypotheses, we establish existence and uniqueness of an entire solution vanishing at infinity. More generally, we discuss existence and uniqueness of an entire solution approaching a given heat polynomial at infinity. When specialized to the linear homogeneous parabolic equation, containing no zero-order term, our results yield a Liouville theorem stating that an entire and bounded solution must be constant; certain asymptotic behavior of the coe cients at infinity however is required. Our methods involve the establishment of a priori bounds on entire solutions, first for the nonhomogeneous heat equation and then for a more general linear parabolic equation; we use these bounds with a Schauder continuation technique to study the nonlinear equation.

Research paper thumbnail of Exceptional boundary sets for solutions of elliptic partial differential inequalities

Indiana University Mathematics Journal, 1986

Etude des ensembles-frontieres exceptionnelles pour les solutions d'inequations aux derivees ... more Etude des ensembles-frontieres exceptionnelles pour les solutions d'inequations aux derivees partielles elliptiques. Generalisation des principes de Phragmen-Lindelof

Research paper thumbnail of Inequalities for eigenvalues of the Laplacian

Indiana University Mathematics Journal, 1980

Research paper thumbnail of Entire solutions of nonlinear parabolic equations

Research paper thumbnail of Higher order Cauchy-Pompeiu operator theory for complex and hypercomplex analysis

Research paper thumbnail of Analytic structure in function algebras

Houston journal of mathematics

Let A be a function algebra with spectrum M and Shilov boundary F. If some function in A is isola... more Let A be a function algebra with spectrum M and Shilov boundary F. If some function in A is isolated-to-one on M-F, then there is an open dense subset U of M-F which is a complex analytic manifold such that every function in A is analytic on U. 1. Introduction. Let X be a compact Hausdorff space, and A a function algebra on X. Let M be the spectrum of A and F its Shilov boundary. Let f G A, and let W be a component of C-fiX]. Bishop [ 5, Section 5 ], Wermer [ 12, Chapter 1 1 ], Aupetit and Wermer [ 1, 2] and Basener [3] have obtained results defining an analytic structure in f'l(w), under the additional assumption that the set f'l(w) = { x G M: f(x)= w} is finite or countable for all w contained in a subset of W of positive measure or capacity. In this paper we obtain analytic structure in all of M-I' under the more restrictive assumption that f is isolated-to-one on M-I'. In particular, we show that if f is isolated-to-one on M-F, then f is an open mapping on M-F, and f is locally a one-to-one mapping (so locally a homeomorphism onto plane domains) except possibly on a nowhere dense subset of M-F. It follows that there is an open dense subset U of M-F which can be given the structure of a one-dimensional complex analytic manifold such that each g in A is analytic on U. There appear to be important differences between our results and those of Bishop, Wermer, Aupetit, and Basener, in particular with regard to the subset of M where analytic structure is.obtained. For example, clearly f'l(w)C M-F for each component W of C-f(X). However, if f(X) meets f(M-I') then t•he collection of sets (f'l(w): W a component ofC-fiX)} does not cover M-F. In fact, for the extreme case when f(X) contains f(M-I-') each *This author was supported in part by grant MCS 76-07180 of the National Science Foundation. 22 H.S. BEAR and G. N. HILE set f-l(w) is empty. Thus we obtain in general an analytic structure on a larger subset of M. On the other hand, while we require that f be isolated-to-one on all of M-F in order to obtain an analytic structure on M-F, in [1], [3], [5], and [12] an analytic structure on f-1 (W) is obtained with the assumption that the inverse images f-1 (w) be finite or countable only on a small portion of W. This latter assumption makes for a relatively more complicated proof, but easier application, as for example to questions of polynomial approximation on an arc in C n. If M is in particular the closure of a plane domain G, we can strengthen our results somewhat. Suppose that F is a proper subset of Uo, and for each z G G-F there is a function fz C A which is countable-to-one in a neighborhood of z. Then there is a homeomorphism Z of G-F onto a plane domain G' such that go Z-1 is analytic on G' for each g C A. Our methods are quite different from those of[1], [2], [3], [5], and [12]. We rely on several purely topological lemmas concerning nearly homeomorphic mappings. These are independent of function algebra assumptions, and are proved in Section 2. Our final theorems (Theorems 3,4) for algebras on a plane domain are extensions of Rudin's work on maximum modulus algebras [8], and our proof uses Rudin's results. We remark that Rudin's results have been extended in a different direction in [4, Section 5]. There it is shown that maximum modulus algebras of functions which satisfy some second order partial differential equations are also algebras of analytic functions of a fixed homeomorphism. 2. Topological Lemmas. We give here some results on nearly homeomorphic maps which do not depend on function algebra assumptions. We will use these facts in the next section. ß Let f be a function on a topological space Y to a topological space Z. We will say that f is n-to-one at y if f-1 (fly)) has n points, and f is countable-to-one at y if f-1 (f(y)) is finite or countably infinite. We say that f is an n-to-one map (or countable-to-one map), if it is n-to-one (countable-to-one) at each point of Y. A map f is finite-to-one (or isolated-to-one)if f-l(f(y)) is a finite set (isolated set) for each y. A function f is locally one-to-one at y if the restriction flu is one-to-one for some neighborhood U of y.

Research paper thumbnail of A Hierarchy of Integral Operators

Rocky Mountain Journal of Mathematics, 1997

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 27, Number 3, Summer 1997 ... Integrals of this type... more ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 27, Number 3, Summer 1997 ... Integrals of this type even in higher dimensions were investigated by Calderon and Zygmund [6, 7]. Because the Π operator for the whole complex plane C turns out to be unitary in ľ(C ...

Research paper thumbnail of Higher order Cauchy Pompeiu operators

Contemporary Mathematics, 1998

Research paper thumbnail of Hilbert function modules with reproducing kernels

Nonlinear Analysis: Theory, Methods & Applications, 1977

Research paper thumbnail of Gradient Characterizations of Analyticity

The American Mathematical Monthly, 1978

... 333 Page 2. 334 HS BEAR AND GN HILE [May Z" U Zc is dense in G, let U be any open subset... more ... 333 Page 2. 334 HS BEAR AND GN HILE [May Z" U Zc is dense in G, let U be any open subset of G. If U n ZC = 0, then U CZ, so U CZ`. ... vx = - bux - cuy ux = bv, + cvy (5) vy = aux + buy; u = -avx - bv. vx = bux +cuy u, = -bvx -cvy (6) vY = - aux - buy; u = avx + bv ...

Research paper thumbnail of Function theory for a class of elliptic systems in the plane

Journal of Differential Equations, 1979

Research paper thumbnail of ftp ejde.math.swt.edu (login: ftp) EXPANSIONS OF SOLUTIONS OF HIGHER ORDER EVOLUTION EQUATIONS IN SERIES OF GENERALIZED HEAT POLYNOMIALS

Abstract. Upper bound estimates are established on generalized heat poly-nomials for higher order... more Abstract. Upper bound estimates are established on generalized heat poly-nomials for higher order linear homogeneous evolution equations with coef-ficients depending on the time variable. These estimates are analogous to well known bounds of Rosenbloom and Widder on the heat polynomials. The bounds lead to further estimates on the width of the strip of convergence of series expansions in terms of these polynomial solutions. An application is given to a Cauchy problem, wherein the solution is expressed as the sum of a series of polynomial solutions. 1.

Research paper thumbnail of ftp ejde.math.swt.edu (login: ftp) HEAT POLYNOMIAL ANALOGS FOR HIGHER ORDER EVOLUTION EQUATIONS

Abstract. Polynomial solutions analogous to the heat polynomials are demon-strated for higher ord... more Abstract. Polynomial solutions analogous to the heat polynomials are demon-strated for higher order linear homogeneous evolution equations with coeffi-cients depending on the time variable. Further parallels with the heat polyno-mials are established when the equation is parabolic with constant coefficients and only highest order terms. 1.

Research paper thumbnail of The behavior of the heat operator on weighted Sobolev spaces

Transactions of the American Mathematical Society, 1998

Denoting by H {\mathcal {H}} the heat operator in R n + 1 R^{n+1} , we investigate its properties... more Denoting by H {\mathcal {H}} the heat operator in R n + 1 R^{n+1} , we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which H {\mathcal {H}} is an injection, a surjection, or an isomorphism. We also describe the range and kernel of H {\mathcal {H}} in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in R n R^{n} .

Research paper thumbnail of Heat Polynomial Analogs

Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, 2003

This study attempts to explore the perceptions of engineering science students (3rd and 4th acade... more This study attempts to explore the perceptions of engineering science students (3rd and 4th academic year) of the exploitation of Information Technology and Communication (ICT) in physics courses. Emphasis is placed exactly on satisfaction of the students next to the use of PowerPoint presentations (PPT), simulations and filmed experiments. To achieve these objectives, we conducted a survey in the form of a questionnaire distributed to 151 students in the engineering cycle in the city of Fez (Morocco). The data collected indicate that 93.4% of students have benefited from the use of Power Point presentations. Among these students only 35.1% consider that these projected slides are a means of facilitating the course content. In addition, 69.5% of students surveyed took advantage of simulations and short sequences filmed during these courses, the majority of them (63.6%) say that these tools help them build their own learning by fostering the intimate link between the course and its application. These results are particularly interesting in the sense that they are used to assess students' perceptions of the simple Power Point presentations. These presentations alone are not enough to make available the contents of a physics course. They must be integrated into pedagogically appropriate learning situations and accompanied by demonstrations on the board. Finally, to illustrate the physical phenomena and physical laws, it is necessary to enrich these slides by simulations and filmed experiments able to develop the cognitive capacities of the student as regards physics.

Research paper thumbnail of Exceptional Boundary Sets for Solutions of Parabolic Partial Differential Inequalities

Transactions of the American Mathematical Society, 1988

Research paper thumbnail of Generalized hypercomplex function theory

Transactions of the American Mathematical Society, 1974

Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distri... more Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distributional solutions of elliptic systems of two equations with two unknowns and two independent variables. Key words and phrases. Hypercomplex variables, pseudo analytic functions of Bers and Vekua, elliptic systems.

Research paper thumbnail of The Cauchy problem and asymptotic decay for solutions of differential inequalities in Hilbert space

Pacific Journal of Mathematics, 1982

Research paper thumbnail of Algebras which satisfy a second order linear partial differential equation

Pacific Journal of Mathematics, 1978

Let A be an algebra of complex valued functions satisfying a second order linear partial differen... more Let A be an algebra of complex valued functions satisfying a second order linear partial differential equation in a plane domain. If the equation is hyperbolic or parabolic, the functions of A are locally functions of only one variable. If the equation is elliptic, there exists a unique complex function Λ such that f x = Λ/ y for each / in A, and after a change of variables each function in A is analytic. If an algebra of functions satisfies the maximum principle, and one nonconstant function and its square satisfy an elliptic equation, then every function in the algebra satisfies this equation.

Research paper thumbnail of Liouville theorems for even order elliptic systems

Indiana University Mathematics Journal

Research paper thumbnail of Liouville theorems for nonlinear parabolic equations of second order

Differential and Integral Equations

We study entire solutions (i.e., solutions defined in all of R n+1) of second-order nonlinear par... more We study entire solutions (i.e., solutions defined in all of R n+1) of second-order nonlinear parabolic equations with linear principal part. Under appropriate hypotheses, we establish existence and uniqueness of an entire solution vanishing at infinity. More generally, we discuss existence and uniqueness of an entire solution approaching a given heat polynomial at infinity. When specialized to the linear homogeneous parabolic equation, containing no zero-order term, our results yield a Liouville theorem stating that an entire and bounded solution must be constant; certain asymptotic behavior of the coe cients at infinity however is required. Our methods involve the establishment of a priori bounds on entire solutions, first for the nonhomogeneous heat equation and then for a more general linear parabolic equation; we use these bounds with a Schauder continuation technique to study the nonlinear equation.

Research paper thumbnail of Exceptional boundary sets for solutions of elliptic partial differential inequalities

Indiana University Mathematics Journal, 1986

Etude des ensembles-frontieres exceptionnelles pour les solutions d'inequations aux derivees ... more Etude des ensembles-frontieres exceptionnelles pour les solutions d'inequations aux derivees partielles elliptiques. Generalisation des principes de Phragmen-Lindelof

Research paper thumbnail of Inequalities for eigenvalues of the Laplacian

Indiana University Mathematics Journal, 1980

Research paper thumbnail of Entire solutions of nonlinear parabolic equations

Research paper thumbnail of Higher order Cauchy-Pompeiu operator theory for complex and hypercomplex analysis

Research paper thumbnail of Analytic structure in function algebras

Houston journal of mathematics

Let A be a function algebra with spectrum M and Shilov boundary F. If some function in A is isola... more Let A be a function algebra with spectrum M and Shilov boundary F. If some function in A is isolated-to-one on M-F, then there is an open dense subset U of M-F which is a complex analytic manifold such that every function in A is analytic on U. 1. Introduction. Let X be a compact Hausdorff space, and A a function algebra on X. Let M be the spectrum of A and F its Shilov boundary. Let f G A, and let W be a component of C-fiX]. Bishop [ 5, Section 5 ], Wermer [ 12, Chapter 1 1 ], Aupetit and Wermer [ 1, 2] and Basener [3] have obtained results defining an analytic structure in f'l(w), under the additional assumption that the set f'l(w) = { x G M: f(x)= w} is finite or countable for all w contained in a subset of W of positive measure or capacity. In this paper we obtain analytic structure in all of M-I' under the more restrictive assumption that f is isolated-to-one on M-I'. In particular, we show that if f is isolated-to-one on M-F, then f is an open mapping on M-F, and f is locally a one-to-one mapping (so locally a homeomorphism onto plane domains) except possibly on a nowhere dense subset of M-F. It follows that there is an open dense subset U of M-F which can be given the structure of a one-dimensional complex analytic manifold such that each g in A is analytic on U. There appear to be important differences between our results and those of Bishop, Wermer, Aupetit, and Basener, in particular with regard to the subset of M where analytic structure is.obtained. For example, clearly f'l(w)C M-F for each component W of C-f(X). However, if f(X) meets f(M-I') then t•he collection of sets (f'l(w): W a component ofC-fiX)} does not cover M-F. In fact, for the extreme case when f(X) contains f(M-I-') each *This author was supported in part by grant MCS 76-07180 of the National Science Foundation. 22 H.S. BEAR and G. N. HILE set f-l(w) is empty. Thus we obtain in general an analytic structure on a larger subset of M. On the other hand, while we require that f be isolated-to-one on all of M-F in order to obtain an analytic structure on M-F, in [1], [3], [5], and [12] an analytic structure on f-1 (W) is obtained with the assumption that the inverse images f-1 (w) be finite or countable only on a small portion of W. This latter assumption makes for a relatively more complicated proof, but easier application, as for example to questions of polynomial approximation on an arc in C n. If M is in particular the closure of a plane domain G, we can strengthen our results somewhat. Suppose that F is a proper subset of Uo, and for each z G G-F there is a function fz C A which is countable-to-one in a neighborhood of z. Then there is a homeomorphism Z of G-F onto a plane domain G' such that go Z-1 is analytic on G' for each g C A. Our methods are quite different from those of[1], [2], [3], [5], and [12]. We rely on several purely topological lemmas concerning nearly homeomorphic mappings. These are independent of function algebra assumptions, and are proved in Section 2. Our final theorems (Theorems 3,4) for algebras on a plane domain are extensions of Rudin's work on maximum modulus algebras [8], and our proof uses Rudin's results. We remark that Rudin's results have been extended in a different direction in [4, Section 5]. There it is shown that maximum modulus algebras of functions which satisfy some second order partial differential equations are also algebras of analytic functions of a fixed homeomorphism. 2. Topological Lemmas. We give here some results on nearly homeomorphic maps which do not depend on function algebra assumptions. We will use these facts in the next section. ß Let f be a function on a topological space Y to a topological space Z. We will say that f is n-to-one at y if f-1 (fly)) has n points, and f is countable-to-one at y if f-1 (f(y)) is finite or countably infinite. We say that f is an n-to-one map (or countable-to-one map), if it is n-to-one (countable-to-one) at each point of Y. A map f is finite-to-one (or isolated-to-one)if f-l(f(y)) is a finite set (isolated set) for each y. A function f is locally one-to-one at y if the restriction flu is one-to-one for some neighborhood U of y.

Research paper thumbnail of A Hierarchy of Integral Operators

Rocky Mountain Journal of Mathematics, 1997

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 27, Number 3, Summer 1997 ... Integrals of this type... more ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 27, Number 3, Summer 1997 ... Integrals of this type even in higher dimensions were investigated by Calderon and Zygmund [6, 7]. Because the Π operator for the whole complex plane C turns out to be unitary in ľ(C ...

Research paper thumbnail of Higher order Cauchy Pompeiu operators

Contemporary Mathematics, 1998

Research paper thumbnail of Hilbert function modules with reproducing kernels

Nonlinear Analysis: Theory, Methods & Applications, 1977

Research paper thumbnail of Gradient Characterizations of Analyticity

The American Mathematical Monthly, 1978

... 333 Page 2. 334 HS BEAR AND GN HILE [May Z" U Zc is dense in G, let U be any open subset... more ... 333 Page 2. 334 HS BEAR AND GN HILE [May Z" U Zc is dense in G, let U be any open subset of G. If U n ZC = 0, then U CZ, so U CZ`. ... vx = - bux - cuy ux = bv, + cvy (5) vy = aux + buy; u = -avx - bv. vx = bux +cuy u, = -bvx -cvy (6) vY = - aux - buy; u = avx + bv ...

Research paper thumbnail of Function theory for a class of elliptic systems in the plane

Journal of Differential Equations, 1979

Research paper thumbnail of ftp ejde.math.swt.edu (login: ftp) EXPANSIONS OF SOLUTIONS OF HIGHER ORDER EVOLUTION EQUATIONS IN SERIES OF GENERALIZED HEAT POLYNOMIALS

Abstract. Upper bound estimates are established on generalized heat poly-nomials for higher order... more Abstract. Upper bound estimates are established on generalized heat poly-nomials for higher order linear homogeneous evolution equations with coef-ficients depending on the time variable. These estimates are analogous to well known bounds of Rosenbloom and Widder on the heat polynomials. The bounds lead to further estimates on the width of the strip of convergence of series expansions in terms of these polynomial solutions. An application is given to a Cauchy problem, wherein the solution is expressed as the sum of a series of polynomial solutions. 1.

Research paper thumbnail of ftp ejde.math.swt.edu (login: ftp) HEAT POLYNOMIAL ANALOGS FOR HIGHER ORDER EVOLUTION EQUATIONS

Abstract. Polynomial solutions analogous to the heat polynomials are demon-strated for higher ord... more Abstract. Polynomial solutions analogous to the heat polynomials are demon-strated for higher order linear homogeneous evolution equations with coeffi-cients depending on the time variable. Further parallels with the heat polyno-mials are established when the equation is parabolic with constant coefficients and only highest order terms. 1.

Research paper thumbnail of The behavior of the heat operator on weighted Sobolev spaces

Transactions of the American Mathematical Society, 1998

Denoting by H {\mathcal {H}} the heat operator in R n + 1 R^{n+1} , we investigate its properties... more Denoting by H {\mathcal {H}} the heat operator in R n + 1 R^{n+1} , we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which H {\mathcal {H}} is an injection, a surjection, or an isomorphism. We also describe the range and kernel of H {\mathcal {H}} in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in R n R^{n} .

Research paper thumbnail of Heat Polynomial Analogs

Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, 2003

This study attempts to explore the perceptions of engineering science students (3rd and 4th acade... more This study attempts to explore the perceptions of engineering science students (3rd and 4th academic year) of the exploitation of Information Technology and Communication (ICT) in physics courses. Emphasis is placed exactly on satisfaction of the students next to the use of PowerPoint presentations (PPT), simulations and filmed experiments. To achieve these objectives, we conducted a survey in the form of a questionnaire distributed to 151 students in the engineering cycle in the city of Fez (Morocco). The data collected indicate that 93.4% of students have benefited from the use of Power Point presentations. Among these students only 35.1% consider that these projected slides are a means of facilitating the course content. In addition, 69.5% of students surveyed took advantage of simulations and short sequences filmed during these courses, the majority of them (63.6%) say that these tools help them build their own learning by fostering the intimate link between the course and its application. These results are particularly interesting in the sense that they are used to assess students' perceptions of the simple Power Point presentations. These presentations alone are not enough to make available the contents of a physics course. They must be integrated into pedagogically appropriate learning situations and accompanied by demonstrations on the board. Finally, to illustrate the physical phenomena and physical laws, it is necessary to enrich these slides by simulations and filmed experiments able to develop the cognitive capacities of the student as regards physics.

Research paper thumbnail of Exceptional Boundary Sets for Solutions of Parabolic Partial Differential Inequalities

Transactions of the American Mathematical Society, 1988

Research paper thumbnail of Generalized hypercomplex function theory

Transactions of the American Mathematical Society, 1974

Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distri... more Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distributional solutions of elliptic systems of two equations with two unknowns and two independent variables. Key words and phrases. Hypercomplex variables, pseudo analytic functions of Bers and Vekua, elliptic systems.

Research paper thumbnail of The Cauchy problem and asymptotic decay for solutions of differential inequalities in Hilbert space

Pacific Journal of Mathematics, 1982

Research paper thumbnail of Algebras which satisfy a second order linear partial differential equation

Pacific Journal of Mathematics, 1978

Let A be an algebra of complex valued functions satisfying a second order linear partial differen... more Let A be an algebra of complex valued functions satisfying a second order linear partial differential equation in a plane domain. If the equation is hyperbolic or parabolic, the functions of A are locally functions of only one variable. If the equation is elliptic, there exists a unique complex function Λ such that f x = Λ/ y for each / in A, and after a change of variables each function in A is analytic. If an algebra of functions satisfies the maximum principle, and one nonconstant function and its square satisfy an elliptic equation, then every function in the algebra satisfies this equation.