parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

power concavity on nonlinear parabolic flows our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p laplace equation called power concavity and that the power is determined by the homogeneity of the parabolic operator in the parabolic p laplace equation for the density u the concavity of u p 2 p is considered which indicates why the log concavity has been considered in heat flow p 2 in addition the long time existence of the classical solution of the parabolic p laplacian equation can be obtained if the initial smooth data has concavity and a nondegenerate gradient along the initial boundary 2004 wiley periodicals inc

power concavity in weakly coupled elliptic and parabolic systems abstract in this paper we start the investigation of concavity properties of solutions to systems of pde s in convex domains in particular we prove that suitable powers of solutions to some weakly coupled elliptic and parabolic systems are concave

finite time generalization of the thermodynamic uncertainty relation for fluctuating currents in nonequilibrium steady states the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving we show that this relation holds not only for the long time limit of fluctuations as described by large deviation theory but also for fluctuations on arbitrary finite time scales this generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments for which infinite time scales are not accessible importantly often this finite time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long time limit we illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle

a microscopic convexity principle for nonlinear partial differential equations we study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations under certain general structure condition we establish that the rank of hessian 2u is of constant rank for any convex solution u of equation f 2u u u x 0 the similar result is also proved for parabolic equations some of geometric applications are also discussed

fluctuation theorem for partially masked nonequilibrium dynamics we establish a generalization of the fluctuation theorem for partially masked nonequilibrium dynamics we introduce a partial entropy production with a subset of all possible transitions and show that the partial entropy production satisfies the integral fluctuation theorem our result reveals the fundamental properties of a broad class of autonomous as well as nonautonomous nanomachines in particular our result gives a unified fluctuation theorem for both autonomous and nonautonomous maxwell s demons where mutual information plays a crucial role furthermore we derive a fluctuation dissipation theorem that relates nonequilibrium stationary current to two kinds of equilibrium fluctuations

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

bound on thermoelectric power in a magnetic field within linear response for thermoelectric power generation in a multiterminal geometry strong numerical evidence for a universal bound as a function of the magnetic field induced asymmetry of the nondiagonal onsager coefficients is presented this bound implies inter alia that the power vanishes at least linearly when the maximal efficiency is approached in particular this result rules out that carnot efficiency can be reached at finite power which an analysis based on the second law only would in principle allow

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

efficiency versus speed in quantum heat engines rigorous constraint from lieb robinson bound a long standing open problem whether a heat engine with finite power achieves the carnot efficiency is investgated we rigorously prove a general trade off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines in a first step employing the lieb robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable this inequality provides a rigorous characterization of the following intuitive picture that most of the energy emitted from the engine to the cold bath remains near the engine when the cyclic process is finished using this description we prove an upper bound on efficiency with the aid of quantum information geometry our result generally excludes the possibility of a process with finite speed at the carnot efficiency in quantum heat engines in particular the obtained constraint covers engines evolving with non markovian dynamics which almost all previous studies on this topic fail to address

carnot cycle at finite power attainability of maximal efficiency we want to understand whether and to what extent the maximal carnot efficiency for heat engines can be reached at a finite power to this end we generalize the carnot cycle so that it is not restricted to slow processes we show that for realistic i e not purposefully designed engine bath interactions the work optimal engine performing the generalized cycle close to the maximal efficiency has a long cycle time and hence vanishing power this aspect is shown to relate to the theory of computational complexity a physical manifestation of the same effect is levinthal s paradox in the protein folding problem the resolution of this paradox for realistic proteins allows to construct engines that can extract at a finite power 40 of the maximally possible work reaching 90 of the maximal efficiency for purposefully designed engine bath interactions the carnot efficiency is achievable at a large power

parabolic power concavity and parabolic boundary value problems this paper is concerned with power concavity properties of the solution to the parabolic boundary value problem begin aligned p quad left begin array l quad l partial t u vardelta u f x t u nabla u text in quad varomega times 0 infty u x t 0 text on quad partial varomega times 0 infty u x 0 0 text in quad varomega end array right end aligned where varomega is a bounded convex domain in mathbf r n and f is a nonnegative continuous function in varomega times 0 infty times mathbf r times mathbf r n we give a sufficient condition for the solution of p to be parabolically power concave in overline varomega times 0 infty

adaptive neural control for output feedback nonlinear systems using a barrier lyapunov function in this brief adaptive neural control is presented for a class of output feedback nonlinear systems in the presence of unknown functions the unknown functions are handled via on line neural network nn control using only output measurements a barrier lyapunov function blf is introduced to address two open and challenging problems in the neuro control area 1 for any initial compact set how to determine a priori the compact superset on which nn approximation is valid and 2 how to ensure that the arguments of the unknown functions remain within the specified compact superset by ensuring boundedness of the blf we actively constrain the argument of the unknown functions to remain within a compact superset such that the nn approximation conditions hold the semiglobal boundedness of all closed loop signals is ensured and the tracking error converges to a neighborhood of zero simulation results demonstrate the effectiveness of the proposed approach

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

attainability of carnot efficiency with autonomous engines the maximum efficiency of autonomous engines with a finite chemical potential difference is investigated we show that without a particular type of singularity autonomous engines cannot attain the carnot efficiency this singularity is realized in two ways single particle transports and the thermodynamic limit we demonstrate that both of these ways actually lead to the carnot efficiency in concrete setups our results clearly illustrate that the singularity plays a crucial role in the maximum efficiency of autonomous engines

smooth solution for the porous medium equation in a bounded domain abstract in this paper we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 u t u m m 1 with zero boundary condition on the fixed boundary the flux u m is nonzero while the gradient of pressure u m 1 is zero as a consequence the parabolic equation above becomes degenerate the proof is based on a definition of weighted space c s 2 corresponding to the given degeneracy and the schauder estimates in its linearized equation 0 2 w t x n a i j x d i j w in c s 2 for 0 m 1 m 1

convexity of level curves for an initial value problem let u be a solution of ut u u7 with dirichlet data equal to 1 on the boundary of a bounded convex domain in rn and initial data also equal to 1 where 0 1 it is shown that u has convex level curves for all time t this problem is used as an example to indicate a general technique for showing that the level curves of solutions of suitable initial value problems are convex

universal trade off relation between power and efficiency for heat engines for a general thermodynamic system described as a markov process we prove a general lower bound for dissipation in terms of the square of the heat current thus establishing that nonvanishing current inevitably implies dissipation this leads to a universal trade off relation between efficiency and power with which we rigorously prove that a heat engine with nonvanishing power never attains the carnot efficiency our theory applies to systems arbitrarily far from equilibrium and does not assume any specific symmetry of the model

proof of the finite time thermodynamic uncertainty relation for steady state currents this paper presents the finite time version of the thermodynamic uncertainty relation for nonequilibrium steady states which sets a bound on the magnitude of current fluctuations the authors use a methodology that is similar to the proof for the long time limit but with large deviation theory applied to an ensemble of many copies

speed limit for classical stochastic processes we consider the speed limit for classical stochastic markov processes with and without the local detailed balance condition we find that for both cases a trade off inequality exists between the speed of the state transformation and the entropy production the dynamical activity is related to a time scale and plays a crucial role in the inequality for the dynamics without the local detailed balance condition we use the hatano sasa entropy production instead of the standard entropy production our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms

hamiltonian derivation of a detailed fluctuation theorem we analyze the microscopic evolution of a system undergoing a far from equilibrium thermodynamic process explicitly accounting for the degrees of freedom of participating heat reservoirs we derive a hybrid result similar in form to both the fluctuation theorem and a statement of detailed balance we relate this result to the steady state fluctuation theorem and to a free energy relation valid far from equilibrium

fundamental relation between entropy production and heat current we investigate the fundamental relation between entropy production rate and the speed of energy exchange between a system and baths in classical markov processes we establish the fact that quick energy exchange inevitably induces large entropy production in a quantitative form more specifically we prove two inequalities on instantaneous quantities one is applicable to general markov processes induced by heat baths and the other is applicable only to systems with the local detailed balance condition but is stronger than the former one we demonstrate the physical meaning of our result by applying to some specific setups in particular we show that our inequality is tight in the linear response regime