existence theorems for positive semidefinite and sign indefinite stabilizing solutions of h riccati equations the existence of both positive semidefinite and sign indefinite stabilizing solutions of h infty type riccati equations is determined via joint eigenvalue and hamiltonian tests these tests determine the inertia properties of the solutions they are dependent on a formula derived in this paper that shows how the solution of game riccati equations can be decomposed into a sum of two well known riccati equations namely the standard filter riccati equation and the bounded real riccati equation other properties of stabilizing solutions such as solution partial order spectral radius monotonicity and rank are also discussed

reconstruction of planar conductivities in subdomains from incomplete data we consider the problem of recovering a sufficiently smooth isotropic conductivity from interior knowledge of the magnitude of the current density field j generated by an imposed voltage potential f on the boundary in any dimension n geq2 we show that equipotential sets are global area minimizers in the conformal metric determined by j in two dimensions assuming the boundary voltage is almost two to one we prove uniqueness of the minimization problem this yields two results on reconstruction from incomplete data in the first case j is known in all of omega but the almost two to one f is know only on subintervals of the boundary the second case assumes that j is known only in an appropriate subdomain tilde omega our method works provided that tilde omega contains entire equipotential curves joining boundary points based on solving two point boundary value problems for the geodesic system we give a procedure to determine whether tilde omega satisfies this pro

least gradient problems with neumann boundary condition abstract we study existence of minimizers of the least gradient problem inf v b v g x d v where b v g v b v g v 1 x p r n r is a convex continuous and homogeneous function of degree 1 with respect to the p variable and g satisfies the compatibility condition g d s 0 we prove that for every 0 g l there are infinitely many minimizers in b v moreover there exists a divergence free vector field t l n that determines the structure of level sets of all minimizers i e t determines d u d u d u a e in for every minimizer u we also prove some existence results for general 1 laplacian type equations with neumann boundary condition a numerical algorithm is presented that simultaneously finds t and a minimizer of the above least gradient problem applications of the results in conductivity imaging are discussed

on the solutions to 1 laplacian equation with l1 data in the present paper we study the behaviour as p goes to 1 of the renormalized solutions to the problems 0 1 div up p 2 up fin up 0on where p 1 is a bounded open set of rn n 2 with lipschitz boundary and f belongs to l1 we prove that these renormalized solutions pointwise converge up to subsequences to a function u with a suitable definition of solution we also prove that u is a solution to a limit problem moreover we analyze the situation occurring when more regular data f are considered

trace theorems for functions of bounded variation in metric spaces abstract in this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1 poincare inequality and obtain l 1 estimates of the trace functions in contrast with the treatment of traces given in other papers on this subject the traces we consider do not require knowledge of the function in the exterior of the domain we also establish a maz ya type inequality for functions of bounded variation that vanish on a set of positive capacity

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

least gradient problem with respect to a non strictly convex norm abstract we study the planar least gradient problem with respect to an anisotropic norm for continuous boundary data we prove existence of minimizers for strictly convex domains furthermore we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of the norm and the domain the results are independent from the regularity of the norm

special cases of the planar least gradient problem abstract we study two special cases of the planar least gradient problem in the first one the boundary conditions are imposed on a part of a strictly convex domain in the second case we impose the dirichlet data on the boundary of a rectangle an example of convex but not strictly convex domain we show the existence of solutions and study their properties for particular cases of boundary data

on weak convergence of the douglas rachford method we prove that the sequences generated by the douglas rachford method converge weakly to zero of the sum of two maximal monotone operators using new tools introduced in recent works of eckstein and the author the assumption of maximal monotonicity of the sum is also removed using a recent result of bauschke

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

continuity of minimizers to weighted least gradient problems abstract we revisit the question of existence and regularity of minimizers to the weighted least gradient problem with dirichlet boundary condition inf a x d u u b v u g where g c and a c 2 is a weight function that is bounded away from zero under suitable geometric conditions on the domain r n we construct continuous solutions of the above problem for any dimension n 2 by extending the sternberg williams ziemer technique sternberg et al 1992 to this setting of inhomogeneous variations we show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a 2 n 1 i n this result complements the approach in jerrard et al 2018 since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities the proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by leon simon in simon 1987

optimality conditions for vector equilibrium problems in this paper we present the necessary and sufficient conditions for weakly efficient solution henig efficient solution globally efficient solution and superefficient solution to the vector equilibrium problems with constraints as applications we give the necessary and sufficient conditions for corresponding solution to the vector variational inequalities and vector optimization problems

scalarization and optimality conditions for vector equilibrium problems abstract in this paper we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions henig efficient solutions and globally efficient solutions to the vector equilibrium problems without the convexity assumption using nonsmooth analysis and the scalarization results we provided the necessary conditions for weakly efficient solutions henig efficient solutions globally efficient solutions and superefficient solutions to vector equilibrium problems by the assumption of convexity we gave sufficient conditions for those solutions as applications we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems

on a degenerate problem in the calculus of variations we establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations the proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets we exploit the uniqueness and the regularity results to explore some of their qualitative properties in particular we emphasize the link between the supports of the solutions and the cheeger problem

a weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging we consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes the required interior data can be obtained by means of mri measurements on the boundary we only require knowledge of the electrodes their impedances and the corresponding average input currents from the mathematical point of view this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the complete electrode model cem of somersalo cheney and isaacson siam j appl math 52 1992 pp 1023 1040 we show that this variational problem has nonunique solutions the surprising discovery is that the physical data is still sufficient to determine the geometry of the connected components of the level sets of the minimizers we thus obtain an interesting phase retrieval result knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude we characterize locally the nonuniqueness in the variational problem in two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity the proofs involve a maximum principle and a new regularity up to the boundary result for the cem boundary conditions a nonlinear algorithm is proposed and implemented to illustrate the theoretical results

on boundary detachment phenomena for the total variation flow with dynamic boundary conditions abstract we combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions we analyze the behavior of facets at the parts of the boundary where these conditions are imposed we devote particular attention to the radially symmetric data we observe that the boundary layer detachment actually can happen at concave parts of the boundary

a new positive definite geometric mean of two positive definite matrices abstract we introduce and study a new positive definite in certain singular cases positive semidefinite geometric mean of two positive definite under certain conditions positive semidefinite matrices

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

the constrained least gradient problem in rn we consider the constrained least gradient problem inf u dx u c 0 1 u 1 a e u g on which arises as the relaxation of a nonconvex problem in optimal design we establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem we also establish the uniqueness of solutions and discuss their structure

on optimality conditions for vector variational inequalities abstract optimality conditions for weak efficient global efficient and efficient solutions of vector variational inequalities with constraints defined by equality cone and set constraints are derived under various constraint qualifications necessary optimality conditions for weak efficient global efficient and efficient solutions in terms of the clarke and michel penot subdifferentials are established with assumptions on quasiconvexity of constraint functions sufficient optimality conditions are also given

congruence of hermitian matrices by hermitian matrices two hermitian matrices a b mn c are said to be hermitian congruent if there exists a nonsingular hermitian matrix c mn c such that b cac in this paper we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable hermitian matrices a and b to be hermitian congruent moreover when a and b are hermitian congruent we describe the possible inertias of the hermitian matrices c that carry the congruence we also give necessary and sufficient conditions for any 2 by 2 nonsingular hermitian matrices to be hermitian congruent in both of the studied cases we show that if a and b are real and hermitian congruent then they are congruent by a real symmetric matrix finally we note that if a and b are 2 by 2 nonsingular real symmetric matrices having the same sign pattern then there is always a real symmetric matrix c satisfying b cac moreover if both matrices are positive then c can be picked with arbitrary inertia