free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

global periodic solutions of the nonlinear wave equation on considere le probleme u tt u t g u t r 2 u t t u t u t t t t t t t ou u et sont des fonctions scalaires de r 2 et g une fonction scalaire sur r on suppose que g et sont continues et on determine des conditions suffisantes sous forme d inegalites pour garantir que ce probleme a des solutions continues

a birkhoff lewis type theorem for the nonlinear wave equation we give an extension of the celebrated birkhoff lewis theorem to the nonlinear wave equation accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin which is an elliptic equilibrium of the associated infinite dimensional hamiltonian system

periodic solutions for the schrodinger equation with nonlocal smoothing nonlinearities in higher dimension abstract we consider the nonlinear schrodinger equation in higher dimension with dirichlet boundary conditions and with a nonlocal smoothing nonlinearity we prove the existence of small amplitude periodic solutions in the fully resonant case we find solutions which at leading order are wave packets in the sense that they continue linear solutions with an arbitrarily large number of resonant modes the main difficulty in the proof consists in a small divisor problem which we solve by using a renormalisation group approach

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

an optimization problem with volume constraint for a degenerate quasilinear operator abstract we consider the optimization problem of minimizing u p d x with a constraint on the volume of u 0 we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

regularity of the free boundary in an optimization problem related to the best sobolev trace constant in this paper we study the regularity properties of a free boundary problem arising in the optimization of the best sobolev trace constant in the immersion h 1 omega hookrightarrow l q partial omega for functions that vanish in a subset of omega this problem is also related to a minimization problem for steklov eigenvalues

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

cantor families of periodic solutions for completely resonant nonlinear wave equations we prove the existence of small amplitude 2 periodic in time solutions of completely resonant nonlinear wave equations with dirichlet boundary conditions for any frequency belonging to a cantor like set of asymptotically full measure and for a new set of nonlinearities the proof relies on a suitable lyapunov schmidt decomposition and a variant of the nash moser implicit function theorem in spite of the complete resonance of the equation we show that we can still reduce the problem to a finitedimensional bifurcation equation moreover a new simple approach for the inversion of the linearized operators required by the nash moser scheme is developed it allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

estimates of the best sobolev constant of the embedding of bv ω into l1 ω and related shape optimization problems abstract in this paper we find estimates for the optimal constant in the critical sobolev trace inequality 1 u l 1 u w 1 1 that are independent of these estimates generalize those of j fernandez bonder n saintier estimates for the sobolev trace constant with critical exponent and applications ann mat pura aplicata in press concerning the p laplacian to the case p 1 we apply our results to prove the existence of an extremal for this embedding we then study an optimal design problem related to 1 and eventually compute the shape derivative of the functional 1

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

a semilinear elliptic problem which is not selfadjoint cesari 7 i in a preceding paper established conditions sufficient that a semilinear elliptic boundary value problem possibly not selfadjoint shall have a solution

three class roc analysis a decision theoretic approach under the ideal observer framework receiver operating characteristic roc analysis is well established in the evaluation of systems involving binary classification tasks however medical tests often require distinguishing among more than two diagnostic alternatives the goal of this work was to develop an roc analysis method for three class classification tasks based on decision theory we developed a method for three class roc analysis in this method the objects were classified by making the decision that provided the maximal utility relative to the other two by making assumptions about the magnitudes of the relative utilities of incorrect decisions we found a decision model that maximized the expected utility of the decisions when using log likelihood ratios as decision variables this decision model consists of a two dimensional decision plane with log likelihood ratios as the axes and a decision structure that separates the plane into three regions moving the decision structure over the decision plane which corresponds to moving the decision threshold in two class roc analysis and computing the true class 1 2 and 3 fractions defined a three class roc surface we have shown that the resulting three class roc surface shares many features with the two class roc curve i e using the log likelihood ratios as the decision variables results in maximal expected utility of the decisions and the optimal operating point for a given diagnostic setting set of relative utilities and disease prevalences lies on the surface the volume under the three class surface vus serves as a figure of merit to evaluate different data acquisition systems or image processing and reconstruction methods when the assumed utility constraints are relevant

optimal boundary holes for the sobolev trace constant in this paper we study the problem of minimizing the sobolev trace rayleigh quotient u w1 p p u lq p among functions that vanish in a set contained on the boundary of given boundary measure we prove existence of extremals for this problem and analyze some particular cases where information about the location of the optimal boundary set can be given moreover we further study the shape derivative of the sobolev trace constant under regular perturbations of the boundary set

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

quasi periodic solutions for fully nonlinear forced reversible schrodinger equations abstract in this paper we consider a class of fully nonlinear forced and reversible schrodinger equations and prove existence and stability of quasi periodic solutions we use a nash moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero due to the presence of the highest order derivatives in the non linearity the classic kam reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo differential operators this procedure automatically produces a change of variables well defined on the phase space of the equation which diagonalizes the operator linearized at the solution this gives the linear stability

a free boundary problem for the p x laplacian abstract we consider the optimization problem of minimizing 1 p x u p x x u 0 d x in the class of functions w 1 p with u 0 w 0 1 p for a given 0 0 and bounded w 1 p is the class of weakly differentiable functions with u p x d x we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface

an optimization problem with volume constraint in orlicz spaces we consider the optimization problem of minimizing g u dx in the class of functions w1 g with a constraint on the volume of u 0 the conditions on the function g allow for a different behavior at 0 and at we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

quasi periodic solutions of a non autonomous wave equations with quasi periodic forcing abstract in this paper we prove existence of small amplitude quasi periodic solutions for a non autonomous quasi periodically forced nonlinear wave equations with periodic spatial boundary conditions via kam theory

standing waves on an infinitely deep perfect fluid under gravity the existence of two dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established when formulated as a second order equation for a real valued function w on the 2 torus and a positive parameter the problem is fully nonlinear the highest order x derivative appears in the nonlinear term but not in the linearization at 0 and completely resonant there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter moreover for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial using a nash moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions the existence question can be reduced to one of estimating the inverses of linearized operators at non zero points after changing coordinates these operators become first order and non local in space and second order in time after further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing or quasi one dimensional in the sense of 22 the operator can then be inverted for two reasons first the explicit formula for the approximate solution means that restricted to the infinite dimensional kernel of the linearization at zero the inverse exists and can be estimated second the small divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their diophantine properties a parameter dependent version of the nash moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth corresponding to a set of values of the parameter 1 which is dense at 1 unimodal means that the term of smallest order in the amplitude is cos x cos t which is one of many eigenfunctions of the completely resonant linearized problem

some nonlocal optimal design problems abstract in this paper we study two optimal design problems associated to fractional sobolev spaces w s p then we find a relationship between these two problems and finally we investigate the convergence when s 1

an optimization problem with volume constraint we consider the optimization problem of minimizing dirichlet integral int omega nabla u 2 with volume constraint on the set u 0 we use a penalization method which for small values of the penalization parameter leads to a solution of the original problem