Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated with the sum of square operator Δx = ∑m(j=1) Xj* Xj.

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Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The first boundary value problem for eΔu + a(x,y,e)ux + b(x,y,e)uy +c(x,y,e)u = d(x,y,e) for small e. This problem with coefficients independent of e was treated by Norman Levinson and appeared in Annals of Mathematics, Vol. 51, No. 2, March, 1950 [TRUNCATED]. / 2999-01-01

Estudamos o Espectro de Fucík para o operador Laplaciano, isto é, o conjunto \'SIGMA\' das duplas (\'mü\', \'nü\') \'ESTA CONTIDO EM\' \'R POT. 2\', tais que o problema { - \'DELTA\' u(x) = \'\'\'mü \'nü\' POT. + (x); \'EPSILON\' \' OMEGA\', Bu = o; x \'EPSILON\' \'PARTIAL\' \' OMEGA\', admita soluções não triviais, onde \'OMEGA \'ESTA CONTIDO EM\' \'R POT. n\' é um domínio limitado, \'u POT +\'(x) = max{0, u(x)}, \' u POT. -\' (x) = f -u (x)} e B representa condições de contorno. Inicialmente apresentamos alguns resultados abstratos sobre o Espectro de Fucík e em seguida o calculamos explicitamente no caso unidimensional para os problemas de Dirichlet e de Neumann. Estes resultados são aplicados ao estudo da solubilidade do problema { - \'DELTA\' u(x) = f(x, u (x)); x \'epsilon\' \'OMEGA\', Bu = 0; x \'epsilon\' \'PARTIAL\' \' OMEGA\', quando a não linearidade f é uma conveniente perturbação de \'mü\'\'u POT. + - \'\'nü\' u+ - \'\'nü\' u POT. n\', descreveremos diferentes comportamentos em função dos parâmetros (\'mü\', \'nü\'). Por fim, consideramos o Espectro de Fucík em dimensão maior. Neste caso não é possível calculá-lo explicitamente, assim apresentamos uma caracterização variacional da sua primeira curva não trivial. Esta caracterização nos permitirá obter várias informações sobre a forma desta curva e também outros resultados sobre a solubilidade de (2) / We study the Fucik Spectrum for the Laplacian operator, that is, the set \'SIGMA\' of the couples (\'mü\', \'nü\') \'ARE THIS ESTA CONTAINED\' \'R POT. 2\', for which the problem { - \'DELTA\' u(x) = \'\'\'mü \'nü\' POT. + (x); \'EPSILON\' \' OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\' \' OMEGA\', admits a nontrivial solution, where \'OMEGA\' \'EPSILON\' \'R POT. n\' is a bounded domain, \'u POT. + (x) = max {0, u(x)}, \'u POT. -\'(x) = {0, - u(x)} and B represents some boundary condition. We first show abstract results about the Fucik Spectrum and then we compute it explicitly in the one dimensional case for the Dirichlet and Neumann problems. These results one applied at the study of the solvability of the problem. { - \'DELTA\'u(x) = f (x, u(x)), x \'EPSILON\' \'OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\'\'OMEGA\', whe3n the nonlinearity f is a suitable pertubation of \'mü\'\'u POT. + - \'\'nü\' u+ - \'\'nü\' u POT. n\'; we describe different behaviors depending on the parameters (\'mü\', \'nü\'). Finally, we consider the Fucik Spectrum in higher dimension. In this case it is not possible to compute it explicitly, so we will show a variational characterization of the first nontrivial curve. This characterization will allow to obtain some information on the properties of this curve and also further results on the solvability of (2)

In this thesis, we investigate Riesz bases of wavelets and their applications to numerical solutions of elliptic equations. Compared with the finite difference and finite element methods, the wavelet method for solving elliptic equations is relatively young but powerful. In the wavelet Galerkin method, the efficiency of the numerical schemes is directly determined by the properties of the wavelet bases. Hence, the construction of Riesz bases of wavelets is crucial. We propose different ways to construct wavelet bases whose stability in Sobolev spaces is then established. An advantage of our approaches is their far superior simplicity over many other known constructions. As a result, the corresponding numerical schemes are easily implemented and efficient. We apply these wavelet bases to solve some important elliptic equations in physics and show their effectiveness numerically. Multilevel algorithm based on preconditioned conjugate gradient algorithm is also developed to significantly improve the numerical performance. Numerical results and comparison with other existing methods are presented to demonstrate the advantages of the wavelet Galerkin method we propose. / Mathematics

The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly $V_h$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly $V_h$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.

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Estudamos o Espectro de Fucík para o operador Laplaciano, isto é, o conjunto \'SIGMA\' das duplas (\'mü\', \'nü\') \'ESTA CONTIDO EM\' \'R POT. 2\', tais que o problema { - \'DELTA\' u(x) = \'\'\'mü \'nü\' POT. + (x); \'EPSILON\' \' OMEGA\', Bu = o; x \'EPSILON\' \'PARTIAL\' \' OMEGA\', admita soluções não triviais, onde \'OMEGA \'ESTA CONTIDO EM\' \'R POT. n\' é um domínio limitado, \'u POT +\'(x) = max{0, u(x)}, \' u POT. -\' (x) = f -u (x)} e B representa condições de contorno. Inicialmente apresentamos alguns resultados abstratos sobre o Espectro de Fucík e em seguida o calculamos explicitamente no caso unidimensional para os problemas de Dirichlet e de Neumann. Estes resultados são aplicados ao estudo da solubilidade do problema { - \'DELTA\' u(x) = f(x, u (x)); x \'epsilon\' \'OMEGA\', Bu = 0; x \'epsilon\' \'PARTIAL\' \' OMEGA\', quando a não linearidade f é uma conveniente perturbação de \'mü\'\'u POT. + - \'\'nü\' u+ - \'\'nü\' u POT. n\', descreveremos diferentes comportamentos em função dos parâmetros (\'mü\', \'nü\'). Por fim, consideramos o Espectro de Fucík em dimensão maior. Neste caso não é possível calculá-lo explicitamente, assim apresentamos uma caracterização variacional da sua primeira curva não trivial. Esta caracterização nos permitirá obter várias informações sobre a forma desta curva e também outros resultados sobre a solubilidade de (2) / We study the Fucik Spectrum for the Laplacian operator, that is, the set \'SIGMA\' of the couples (\'mü\', \'nü\') \'ARE THIS ESTA CONTAINED\' \'R POT. 2\', for which the problem { - \'DELTA\' u(x) = \'\'\'mü \'nü\' POT. + (x); \'EPSILON\' \' OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\' \' OMEGA\', admits a nontrivial solution, where \'OMEGA\' \'EPSILON\' \'R POT. n\' is a bounded domain, \'u POT. + (x) = max {0, u(x)}, \'u POT. -\'(x) = {0, - u(x)} and B represents some boundary condition. We first show abstract results about the Fucik Spectrum and then we compute it explicitly in the one dimensional case for the Dirichlet and Neumann problems. These results one applied at the study of the solvability of the problem. { - \'DELTA\'u(x) = f (x, u(x)), x \'EPSILON\' \'OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\'\'OMEGA\', whe3n the nonlinearity f is a suitable pertubation of \'mü\'\'u POT. + - \'\'nü\' u+ - \'\'nü\' u POT. n\'; we describe different behaviors depending on the parameters (\'mü\', \'nü\'). Finally, we consider the Fucik Spectrum in higher dimension. In this case it is not possible to compute it explicitly, so we will show a variational characterization of the first nontrivial curve. This characterization will allow to obtain some information on the properties of this curve and also further results on the solvability of (2)

In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough "mass " is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem. Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all of which produce new results. Our method allows the construction of solutions with prescribed symmetry, and it represents a step further in the classification of positive solutions for certain nonlinear elliptic problems.

We are concerned with the so-called asymptotic Plateau problem in hyperbolic space. That is, to prove the existence of hypersurfaces in hyperbolic space whose principal curvatures satisfy a general curvature relation and has a precribed asymptotic boundary at infinity. In this thesis, by following the method of Bo Guan, Joel Spruck and their collaborators, we solve the problem with the aid of an additional assumption. In particular, our result applies to hypersurfaces whose principal curvatures lie in the k-th Garding cone and has constant (k,k-1) curvature quotient. / Thesis / Master of Science (MSc)