We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey.

We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.

This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.

We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.

We study elliptic boundary value problems in a wedge with additional edge conditions of trace and potential type. We compute the (difference of the) number of such conditions in terms of the Fredholm index of the principal edge symbol. The task will be reduced to the case of special opening angles, together with a homotopy argument.

Neste trabalho estudaremos a geração do semigrupos por operadores elípticos em dois espaços. Em primeiro lugar estudaremos a geração de semigrupo no espaço \'L POT.2\' (\'OMEGA\') por operadores elípticos de ordem 2m com \'OMEGA\' suficientemente regular. Mais precisamente, se \'OMEGA\' é um domínio limitado com \'PARTIAL OMEGA\' de classe \'C POT. 2m,\' L (x;D) = \'SIGMA\' / [\'alpha\'] \'< ou =\' \'a IND. alpha\' (x) \'D POT. alpha\' é um operador diferencial elíptico de ordem 2m, com \'a IND. alpha\' \'PERTENCE\' \' \'C POT.j\' (\'OMEGA\'), j = max {0, [\'alpha\'] - m}, e A : D(A) \'ESTÁ CONTIDO\' EM \'L POT. 2 (\'OMEGA\') \'SETA\' \' L POT. 2 (\'OMEGA\') é o operador linear dado por D(A) = \'H POT. 2m\' (\'OMEGA\') \'H POT. m INF. 0\' (\'OMEGA\'), (Au)(x) = L (x;D)u; então -A gera um \'C IND. 0\'-semigrupo holomorfo em \'L POT.2\' (\'OMEGA\'). ). Em segundo lugar estudaremos a geração de semigrupo em \'C IND. 0\'(\'OMEGA\") = ) = {u \'PERTENCE A\' C (\'OMEGA\' \'BARRA\") : u[\'PARTIAL omega\' = 0} por operadores elípticos de ordem 2 com \'OMEGA\' satisfazendo uma propriedade geométrica. Mais precisamente, se \'OMEGA\' ESTA CONTIDO EM\' \'R POT. n\' (n \'> ou =\' 2) é um domínio limitado que satisfaz a condição de cone exterior uniforme, L é o operador Lu := - \\\\SIGMA SUP n INF. i,j = 1\' \'a IND. ij \'D IND. ij u + \'\\SIGMA SUP. n IND. j=1 \'b IND. j\' u + cu com coeficientes reais \'a IND. ij\' , \'b IND. j\' , c que satisfazem \'b IND. j \' \'PERTENCE A\' \'L POT. INFTY\' (\'OMEGA\') , j = 1, ..., n, c \'PERTENCE A \' \'L POT> INFTY\' (OMEGA), c \'> ou =\' 0, \'a IND. ij\' \'PERTECE A\' C(\' OMEGA BARRA)\' \' INTERSECCAO\' \'L POT. INFTY\' (OMEGA),e \'A IND. 0\' é parte de L em \'C IND. 0\' (\"OMEGA\'), isto é, D(\'A IND. 0\') = {u \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\') \'INTERSECÇÂO\' \'W POT. 2, n INF. loc\' (\'OMEGA\') : Lu \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\')\' \'A IND. 0\' u = Lu, então -\'A IND. 0\' gera um \'C IND. 0-semigrupo holomorfo limitado em \'C IND. 0\' (\'OMEGA\') / In this work we study the generation of semigroups by elliptic operators in two spaces. Firstly we study the generation of semigroup in the space \'L POT. 2\' (OMEGA) for elliptic operators of order 2m with \'OMEGA\' regular domain. More precisely, if \'OMEGA\' is a bounded domain with \\PARTIAL OMEGA\' \'IT BELONGS\' \'C POT. 2m\', L (x, D) = \\ sigma INF.ALPHA \'> or =\' 2m, \'a IND. alpha\' ( x) \'D POT alpha\' is an elliptic differential operator of order 2m, with \'a IND. alpha\' \' \'IT BELONGS\' \'C POT. j\' (OMEGA), j = max , and A : D (A) \'THIS CONTAINED\' \'L POT. 2\' (OMEGA) \'ARROW\' \'L POT. 2\' (OMEGA) is linear operator given or D(A) = \'H POT. 2m\' (OMEGA) \'INTERSECTION\' \'H POT. m INF. 0 (OMEGA) (Au) (x) = L (x,D) u then -A generates a holomorphic \'C IND. 0\'-semigroup in \'L POT. 2\'.(OMEGA). Secondly we study the generation of semigroup in \'C IND. 0\' (OMEGA) = {u \'IT BELONGS\' (c INF. O\' (OMEGA BAR) : \'u [IND. \\partial omega\' = 0} for elliptic operators of second order with \'OMEGA\' satisfying a geometric property. That is, if \'OMEGA\' \'IT BELONGS\' \'R POT. n\' (n > or = 2) is a bounded domain that satisfies the uniform exterior cone condition, L is the elliptic operator given by Lu : = - \\SIGMA SUP. n INF. i,j = 1\' \'a IND. i, j\' \'D IND. ij \' u + \\SIGMA SUP n INF. j=1\' \'b IND j D IND j\' u + cu with real coefficients \'a IND. ij, \'b IND. j\' , c satisfying \'b ind. j\' \'IT BELONGS\' \' L POT. INFTY\' (omega), j = 1, ..., n, c \'it belongs\' \'L POT. INFTY\' (OMEGA), \'c > or =\' 0, \'\'a IND. ij \'IT BELONGS\' C (OMNEGA BAR) \'INTERSECTION\' (OMEGA), and \'A IND. 0\' is part of L in \'C IND. 0\'(OMEGA), that is, D (\'A IND. 0\') = {u \'IT BELONGS\' \'C IND. 0\' (OMEGA) INTERSECTION \'W POT. 2, n IND. loc (OMEGA)} \'A IND. 0u\' = Lu, then - \'A IND. 0\' generates a bounded holomorphic \'C IND. 0\'-semigroup on \'C IND. 0\' (OMEGA)

Neste trabalho estudaremos a geração do semigrupos por operadores elípticos em dois espaços. Em primeiro lugar estudaremos a geração de semigrupo no espaço \'L POT.2\' (\'OMEGA\') por operadores elípticos de ordem 2m com \'OMEGA\' suficientemente regular. Mais precisamente, se \'OMEGA\' é um domínio limitado com \'PARTIAL OMEGA\' de classe \'C POT. 2m,\' L (x;D) = \'SIGMA\' / [\'alpha\'] \'< ou =\' \'a IND. alpha\' (x) \'D POT. alpha\' é um operador diferencial elíptico de ordem 2m, com \'a IND. alpha\' \'PERTENCE\' \' \'C POT.j\' (\'OMEGA\'), j = max {0, [\'alpha\'] - m}, e A : D(A) \'ESTÁ CONTIDO\' EM \'L POT. 2 (\'OMEGA\') \'SETA\' \' L POT. 2 (\'OMEGA\') é o operador linear dado por D(A) = \'H POT. 2m\' (\'OMEGA\') \'H POT. m INF. 0\' (\'OMEGA\'), (Au)(x) = L (x;D)u; então -A gera um \'C IND. 0\'-semigrupo holomorfo em \'L POT.2\' (\'OMEGA\'). ). Em segundo lugar estudaremos a geração de semigrupo em \'C IND. 0\'(\'OMEGA\") = ) = {u \'PERTENCE A\' C (\'OMEGA\' \'BARRA\") : u[\'PARTIAL omega\' = 0} por operadores elípticos de ordem 2 com \'OMEGA\' satisfazendo uma propriedade geométrica. Mais precisamente, se \'OMEGA\' ESTA CONTIDO EM\' \'R POT. n\' (n \'> ou =\' 2) é um domínio limitado que satisfaz a condição de cone exterior uniforme, L é o operador Lu := - \\\\SIGMA SUP n INF. i,j = 1\' \'a IND. ij \'D IND. ij u + \'\\SIGMA SUP. n IND. j=1 \'b IND. j\' u + cu com coeficientes reais \'a IND. ij\' , \'b IND. j\' , c que satisfazem \'b IND. j \' \'PERTENCE A\' \'L POT. INFTY\' (\'OMEGA\') , j = 1, ..., n, c \'PERTENCE A \' \'L POT> INFTY\' (OMEGA), c \'> ou =\' 0, \'a IND. ij\' \'PERTECE A\' C(\' OMEGA BARRA)\' \' INTERSECCAO\' \'L POT. INFTY\' (OMEGA),e \'A IND. 0\' é parte de L em \'C IND. 0\' (\"OMEGA\'), isto é, D(\'A IND. 0\') = {u \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\') \'INTERSECÇÂO\' \'W POT. 2, n INF. loc\' (\'OMEGA\') : Lu \'PERTENCE A\' \'C IND. 0\' (\'OMEGA\')\' \'A IND. 0\' u = Lu, então -\'A IND. 0\' gera um \'C IND. 0-semigrupo holomorfo limitado em \'C IND. 0\' (\'OMEGA\') / In this work we study the generation of semigroups by elliptic operators in two spaces. Firstly we study the generation of semigroup in the space \'L POT. 2\' (OMEGA) for elliptic operators of order 2m with \'OMEGA\' regular domain. More precisely, if \'OMEGA\' is a bounded domain with \\PARTIAL OMEGA\' \'IT BELONGS\' \'C POT. 2m\', L (x, D) = \\ sigma INF.ALPHA \'> or =\' 2m, \'a IND. alpha\' ( x) \'D POT alpha\' is an elliptic differential operator of order 2m, with \'a IND. alpha\' \' \'IT BELONGS\' \'C POT. j\' (OMEGA), j = max , and A : D (A) \'THIS CONTAINED\' \'L POT. 2\' (OMEGA) \'ARROW\' \'L POT. 2\' (OMEGA) is linear operator given or D(A) = \'H POT. 2m\' (OMEGA) \'INTERSECTION\' \'H POT. m INF. 0 (OMEGA) (Au) (x) = L (x,D) u then -A generates a holomorphic \'C IND. 0\'-semigroup in \'L POT. 2\'.(OMEGA). Secondly we study the generation of semigroup in \'C IND. 0\' (OMEGA) = {u \'IT BELONGS\' (c INF. O\' (OMEGA BAR) : \'u [IND. \\partial omega\' = 0} for elliptic operators of second order with \'OMEGA\' satisfying a geometric property. That is, if \'OMEGA\' \'IT BELONGS\' \'R POT. n\' (n > or = 2) is a bounded domain that satisfies the uniform exterior cone condition, L is the elliptic operator given by Lu : = - \\SIGMA SUP. n INF. i,j = 1\' \'a IND. i, j\' \'D IND. ij \' u + \\SIGMA SUP n INF. j=1\' \'b IND j D IND j\' u + cu with real coefficients \'a IND. ij, \'b IND. j\' , c satisfying \'b ind. j\' \'IT BELONGS\' \' L POT. INFTY\' (omega), j = 1, ..., n, c \'it belongs\' \'L POT. INFTY\' (OMEGA), \'c > or =\' 0, \'\'a IND. ij \'IT BELONGS\' C (OMNEGA BAR) \'INTERSECTION\' (OMEGA), and \'A IND. 0\' is part of L in \'C IND. 0\'(OMEGA), that is, D (\'A IND. 0\') = {u \'IT BELONGS\' \'C IND. 0\' (OMEGA) INTERSECTION \'W POT. 2, n IND. loc (OMEGA)} \'A IND. 0u\' = Lu, then - \'A IND. 0\' generates a bounded holomorphic \'C IND. 0\'-semigroup on \'C IND. 0\' (OMEGA)

Les travaux réalisés dans cette thèse concernent l’étude de problèmes quasi-linéaires paraboliques et elliptiques singuliers. Par singularité, nous signifions que le problème fait intervenir une non linéarité qui explose au bord du domaine où l’équation est posée. La présence du terme singulier entraine un manque de régularité des solutions. Ce défaut de régularité génère en conséquence un manque de compacité qui ne permet pas d’appliquer directement les méthodes classiques d’analyse non linéaires pour démontrer l’existence de solutions et discuter les propriétés de régularité et de comportement asymptotique des solutions. Pour contourner cette difficulté dans le contexte des problèmes que nous avons étudiés, nous sommes amenés à établir des estimations a priori très fines au voisinage du bord en combinant diverses méthodes : méthodes de monotonie (reliées au principe du maximum), méthodes variationnelles, argument de convexité, méthodes d’interpolation dans les espaces de Sobolev, méthodes de point fixe. / In this thesis I have studied the Evolution p-laplacian equation with singular nonlinearity. We start by studying the corresponding elliptic problem and then by defining a proper cone in a suitable Sobolev space find the uniqueness of the solution. Taking that into account and using the semi discretization in time we arrive at the uniqueness and existence result. Next we prove some regularity theorem using tools from Nonlinear Semigroup theory and Interpolation spaces. We also establish some related result for the laplacian case where we improve our result on the existence and regularity, due to the non degeneracy of the laplacian. In another related work we work with a semilinear equation with singular nonlinearity and using the moving plane method prove the symmetry properties of any classical solution. We also give some related apriori estimates which together with the symmetry provide us the existence of solution using the bifurcation result.

In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.

The Cayley-Hamilton Theorem is an important result in the study of linear transformations over finite dimensional vector spaces. In this thesis, we show that the Cayley-Hamilton Theorem can be extended to self-adjoint trace-class operators and to closed self-adjoint operators with trace-class resolvent over a separable Hilbert space. Applications of these results include calculating operators resolvents and finding the inverse of a frame operator.