maximum principles and singular elliptic inequalities abstract in this paper we present a version of the omori yau maximum principle a liouville type result and a phragmen lindeloff type theorem for a class of singular elliptic operators on a riemannian manifold which include the p laplacian and the mean curvature operator some applications of the results obtained are discussed

strong maximum principle for general quasilinear problems in non divergence form abstract let us consider a solution u of the following i j a i j x u u 2 u x i x j f x u u in r n u x 0 x r n in this paper we give i some conditions ensuring a strong maximum principle for u ii some conditions ensuring that u has a compact support

thermoacoustic tomography for an integro differential wave equation modeling attenuation abstract in this article we study the inverse problem of thermoacoustic tomography tat on a medium with attenuation represented by a time convolution or memory term and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters under the assumption of being able to measure data on the whole boundary we prove uniqueness and stability and propose a convergent reconstruction method for a class of smooth variable sound speeds by a suitable modification of the time reversal technique we obtain a neumann series reconstruction formula

a note on the strong maximum principle and the compact support principle abstract in this note we are concerned with the strong maximum principle smp and the compact support principle csp for non negative solutions to quasilinear elliptic inequalities of the form div a u u g u f u 0 in and div a u u g u f u 0 in r n b r 0 respectively we give new conditions on the data a g f to obtain smp and csp when these conditions are particularized to the m laplacian and pure power nonlinearities we completely classify the data according to the validity of the csp or the smp in doing so we clarify the general situation and we consider a case not covered in the literature

inversion of spherical means and the wave equation in even dimensions we establish inversion formulas of the so called filtered back projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball we also find several formulas to recover initial data of the form f 0 or 0 g for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball provided that the initial data has support in the ball

results for degenerate nonlinear elliptic equations involving a hardy potential abstract consider the problem p div a x u u u s x p f x in u x 0 in u x 0 on where is an open bounded subset of r n n 3 1 p n and s are positive numbers f is a nonnegative function in some lebesgue space a r r n r n is such that c 0 a x t p 1 p a x t for some 0 1 which provides a noncoercive operator when u the problem could be seen as a reaction model which produces a saturation effect that is the diffusion goes to zero when u goes to infinity this type of reaction appears as a linearization of the arrhenius reaction in some solid combustion problems see for instance peral and vazquez 1995 21 the aim of the article is to study existence nonexistence and regularity of the solutions to problem p the main features in the paper are the following the threshold on s to have existence or nonexistence of solution in some weak sense is s p 1 1 moreover a bound on is needed to have solution in the critical case s p 1 1 the regularity of the solution according to the summability of f

recovering functions from the spherical mean transform with limited radii data by expansion into spherical harmonics abstract the aim of the article is to generalize the method presented in 3 theorem 1 by g ambartsoumian r gouia zarrad and m lewis for recovering functions from their spherical mean transform with limited radii data from the two dimensional case to the general n dimensional case the idea behind the method is to expand each function in question into spherical harmonics and then obtain for each term in the expansion an integral equation of volterra s type that can be solved iteratively we show also how this method can be modified for the spherical case of recovering functions from the spherical transform with limited radii data lastly we solve the analogous problem for the case of the funk transform by again using expansion into spherical harmonics and then obtain an abel type integral equation which can be inverted by a method introduced in 14

existence of positive solutions for nonlinear elliptic equations with convection terms abstract we prove the existence of positive solutions for the equation i 1 n x i a i x u u f x u u in under the dirichlet boundary condition where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient u we develop an approach based on approximate solutions and on a new strong maximum principle

combined effects in nonlinear problems arising in the study ofanisotropic continuous media the paper deals with the study of a lane emden fowler equation with dirichlet boundary condition and variable potential functions the analysis developed in this paper combines monotonicity methods with variational arguments

global attractors and convergence to equilibrium for degenerate ginzburg landau and parabolic equations abstract we study the dynamics of a degenerate complex ginzburg landau cgl and a real parabolic equation with a variable generally nonsmooth diffusion coefficient which may vanish at some points or be unbounded for the cgl equation there exists a global attractor in l 2 for the parabolic equation we show the existence of a global branch of nonnegative stationary states the global bifurcation result is used in order to establish in conjunction with the definition of a gradient dynamical system in the natural phase space that any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium

local vanishing properties of solutions of elliptic and parabolic quasilinear equations this paper is a study of some vanishing properties of weak solutions to nonlinear elliptic and parabolic equations instead of using monotonicity arguments the method of proof is based on an energy method

some criteria for the existence of limit cycles for quadratic vector fields abstract in this paper we consider real quadratic systems we present new criteria for the existence and uniqueness of limit cycles for such systems by using darbouxian particular solutions some results are based on the study of such systems in cp 2 we also generalize the well know result of bautin on the nonexistence of limit cycles for quadratic lotka volterra systems

a mollification approach for inverting the spherical mean radon transform the filtered backprojection algorithm is still one of the most important reconstruction methods in computed tomography it is based on explicit formulas for recovering mollified versions of the original unknown function in this paper we derive such formulas for the spherical mean radon transform with centers restricted to the boundary of a ball in two spatial dimensions we present an efficient implementation strategy that leads to a stable reconstruction algorithm of filtered backprojection type

inversion of circular means and the wave equation on convex planar domains we study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary w of a smooth convex bounded domain w r 2 as a main result we establish back projection type inversion formulas that recover any initial data with support in w modulo an explicitly computed smoothing integral operator k w for circular and elliptical domains the operator k w is shown to vanish identically and hence we establish exact inversion formulas of the back projection type in these cases similar results are obtained for recovering a function from its mean values over circles with centers on w both reconstruction problems are amongst others essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography

sampling conditions for the circular radon transform recovering a function from circular or spherical mean values is the basis of many modern imaging technologies such as photo and thermoacoustic computed tomography or ultrasound reflection tomography recently much progress has been made concerning the problem of recovering a function from its circular mean values its circular radon transform in particular theoretically exact inversion formulas of the back projection type have been discovered using continuously sampled data in practical applications however only a discrete number of circular mean values can be collected in this paper we address this issue in the context of the shannon sampling theory we derive sharp sampling conditions for the number of angular and radial samples such that any essentially b0 bandlimited function can be recovered from a finite number of such circular mean values

the universal back projection formula for spherical means and the wave equation on certain quadric hypersurfaces abstract in many tomographic applications and elsewhere there arises a need to reconstruct a function from the data on the boundary of some domain given either by the spherical means of the function or by the corresponding solution of the free space wave equation in this paper we show that the so called universal back projection formulas provide exact recovery of the unknown function with compact support for data on any quadric hypersurfaces that can be approximated by elliptic hypersurfaces these quadric hypersurfaces include elliptic paraboloid as well as parabolic and elliptic cylinders

inversion formulas and stability estimates of the wave operator on the hyperplane abstract one of the mathematical problems arising in photoacoustic tomography pat a novel and promising technology in medical imaging is to recover the initial function from the solution of the wave equation on a surface where the detectors of pat are located we define the wave operator as the transform assigning to a given function f the solution of the wave equation on the detector surface where the detectors are located with the initial function f here we study many properties of this wave operator including the inversion formulas and stability estimates assuming that the detector surface is a hyperplane

is a curved flight path in sar better than a straight one in the plane we study the transform r gamma f of integrating an unknown function f over circles centered at a given curve gamma this is a simplified model of synthetic aperture radar sar when the radar is not directed but has other applications like thermoacoustic tomography for example we study the problem of recovering the wave front set wf f if the visible singularities of f hit gamma once we show that wf f cannot be recovered i e the artifacts cannot be resolved if gamma partial omega is the boundary of a strictly convex domain omega we show that this is still true on the other hand in the latter case if f is known a priori to have singularities in a compact set then we show that one can recover wf f omega and moreover this can be done in a simple explicit way using backpropagation for the wave equation

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

an inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions abstract in the articles 1 and 2 of d finch m haltmeier s patch and d rakesh inversion formulas were found in any dimension greater than or equal to 2 for recovering a smooth function with compact support in the unit ball from spherical means centered on the unit sphere the aim of this article is to show that the methods used in 1 2 can be modified in order to get similar inversion formulas from spherical means centered on an ellipsoid in two and three dimensional spaces

to local reconstruction from the spherical mean radon transform abstract the article suggests a new approach using the so called consistency method for the inversion of the spherical mean radon transform in 2d with detectors on a line we present a new iterative formula which gives an algorithm to recover an unknown function supported completely on one side of a line l from its spherical means over circles centered on the line l our reconstruction formula has the benefit of being local such an inversion is required in problems of thermo and photo acoustic tomography ultrasound reflection tomography radar imaging and a few other

bayesian tracking and parameter learning for non linear multiple target tracking models this paper proposes a new bayesian tracking and parameter learning algorithm for non linear and non gaussian multiple target tracking mtt models a markov chain monte carlo mcmc algorithm is designed to sample from the posterior distribution of the target states birth and death times and association of observations to targets which constitutes the solution to the tracking problem as well as the model parameters the numerical section presents performance comparisons with several competing techniques and demonstrates significant performance improvements in all cases

kirchhoff type problems on a geodesic ball of the hyperbolic space abstract in this paper we study the existence of weak solutions for some kirchhoff type problems whose simple prototype is given by a b b h u 2 d h u f u in b r u 0 on b r where h denotes the laplace beltrami operator on the ball model of the hyperbolic space b n with n 3 a b and are real parameters b r b n is a geodesic ball centered in zero of radius r and f is a subcritical continuous function the kirchhoff term is allowed to vanish at the origin covering the degenerate case the main technical approach is based on variational and topological methods

the spherical radon transform with centers on cylindrical surfaces abstract recovering a function from its spherical radon transform with centers of spheres of integration restricted to a hypersurface is at the heart of several modern imaging technologies including sar ultrasound imaging and photo and thermoacoustic tomography in this paper we study an inversion of the spherical radon transform with centers of integration restricted to cylindrical surfaces of the form r m where is a hypersurface in r n we show that this transform can be decomposed into two lower dimensional spherical radon transforms one with centers on and one with a planar center set in r m 1 together with explicit inversion formulas for the spherical radon transform with a planar center set and existing algorithms for inverting the spherical radon transform with a center set this yields reconstruction procedures for general cylindrical domains in the special case of spherical or elliptical cylinders we obtain novel explicit inversion formulas for three spatial dimensions these inversion formulas can be implemented efficiently by backprojection type algorithms only requiring o n 4 3 floating point operations where n is the total number of unknowns to be recovered we present numerical results demonstrating the efficiency of the derived algorithms

estimates on the support of the solutions of some nonlinear elliptic and parabolic problems we obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one dimensional motion of a non newtonian fluid precise estimates for the support of these solutions are obtained and the optimality of our hypotheses is discussed

a galerkin least squares approach for photoacoustic tomography the development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography in this paper we address this issue for photoacoustic computed tomography in circular geometry we investigate the galerkin least squares method for that purpose for approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function this includes many systems that have previously been employed in photoacoustic tomography such as generalized kaiser bessel basis functions or the natural pixel basis by exploiting an isometry property of the forward problem we are able to efficiently set up the galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution we establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm

existence of global positive solutions of semilinear elliptic equations abstract using a very simple approach we prove the existence of asymptotically vanishing global positive solutions of the semilinear elliptic equation u f x u g x x u 0 in all of r n n 3 in particular earlier conditions on the negative part of the function f are dropped

keller osserman type conditions for differential inequalities with gradient terms on the heisenberg group abstract we study the qualitative behavior of non negative entire solutions of differential inequalities with gradient terms on the heisenberg group we focus on two classes of inequalities u f u l u and u f u h u g u where f l h g are non negative continuous functions satisfying certain monotonicity properties the operator called the laplacian generalizes the p laplace operator considered by various authors in this setting we prove some liouville theorems introducing two new keller osserman type conditions both extending the classical one which appeared long ago in the study of the prototype differential inequality u f u in r m we show sharpness of our conditions when we specialize to the p laplacian while proving these results we obtain a strong maximum principle for which to the best of our knowledge seems to be new our results continue to hold with the obvious minor modifications also for euclidean space

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

range descriptions for the spherical mean radon transform abstract the transform considered in the paper averages a function supported in a ball in r n over all spheres centered at the boundary of the ball this radon type transform arises in several contemporary applications e g in thermoacoustic tomography and sonar and radar imaging range descriptions for such transforms are important in all these areas for instance when dealing with incomplete data error correction and other issues four different types of complete range descriptions are provided some of which also suggest inversion procedures necessity of three of these appropriately formulated conditions holds also in general domains while the complete discussion of the case of general domains would require another publication

effective discretization of direct reconstruction schemes for photoacoustic imaging in spherical geometries recently kernel methods for the recovery of a function from its spherical means in spherical acquisition geometry have been proposed we present efficient algorithms for these formulas in the two and three dimensional cases our scheme applies fourier techniques for certain convolution type integrals and discretizes physical space on a polar and a spherical grid respectively