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This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.

The set-theoretical convergence of models of continuous and discrete synthesis processes in the compositions of metals and non-metals at the action of ray sources is shown. The non-metal density function obtained by the IDS thermal impulses on the temperature axis projection has a periodic view with fi xed stationary temperatures. They coincide with the boundaries of diff erent areas of change in the speeds of metal heating. This convergence of models to the general periodic operator is possible under four conditions of beam scanning: separation, compactness, property of states, stability. Then the current reciprocity ratios of the system (metals/non-metals) allow the use of the temperature of stationary non-metals as support for metals and serves to determine the parameters of ray sources when synthesizing structures in the compositions of the volume of the material.

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the $C^2$-compactness for all $4$-manifolds (which may be non-umbilic). For the $5$-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the $C^2$-compactness for all $5$-manifolds. Finally, we show that the $C^2$-compactness on $6$-manifolds is true if the trace-free second fundamental form on the boundary never vanishes., Comment: 29 pages, This version treats general 5-manifolds as well, Comments are welcome

We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump L\'evy processes with some $\sigma$-stable like behaviour. Included are $\sigma$-stable processes and fractional Laplace diffusion operators $(-\Delta)^{\frac{\sigma}2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $\sigma\in(1,2)$. We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs uses pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques requires no moment assumptions and uses a weaker topology than Wasserstein., Comment: Major update. A much larger class of anomalous diffusions, local coupling results are now complete, problems are posed in the whole space and not the compact torus. New equicontinuity, $L^1$, and $L^\infty$-results for the Fokker-planck equation are needed and proved by using pure PDE arguments. We now work in a more general metrics than Wasserstein

In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119 , 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.

This paper focuses on the analysis of static spherically symmetric anisotropic solutions in the presence of electromagnetic field through the gravitational decoupling approach in f ( R , T , R γ υ T γ υ ) gravity. We use geometric deformation only on radial metric function and obtain two sets of the field equations. The first set deals with isotropic fluid while the second set yields the influence of anisotropic source. We consider the modified Krori–Barua charged isotropic solution for spherical self-gravitating star to deal with the isotropic system. The second set of the field equations is solved by taking two different constraints. We then investigate physical acceptability of the obtained solutions through graphical analysis of the effective physical variables and energy conditions. We also analyze the effects of charge on different parameters, (i.e., mass, compactness and redshift) for the resulting solutions. It is found that our both solutions are viable as well as stable for specific values of the decoupling parameter φ and charge. We conclude that a self-gravitating star shows more stable behavior in this gravity.

We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T . We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in H p s ( T ) , with s ≥ 0 . The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32] .

We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty $. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty $. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sjöstrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.

We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in $$\mathbb R^4$$ involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is $$2^*=4$$ , there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to $$\mathbb R^N$$ of the Struwe’s global compactness theorem.

In this paper, we consider the asymptotic behavior of solutions to stochastic strongly damped wave equations with variable delays on unbounded domains, which is driven by both additive noise and deterministic non-autonomous forcing. We first establish a continuous cocycle for the equations. Then we prove asymptotic compactness of the cocycle by tail-estimates and a decomposition technique of solutions. Finally, we obtain the existence of a tempered pullback random attractor.

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnstr\"om and Wahl\'en. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve., Comment: Journal version. Mathematische Annalen, Online First. 45 pages, 3 figures

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Namely, we study the unique solvability of the mixed biharmonic problem with the Steklov and Steklov-type conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight $$|x|^{a}$$ . For solving this biharmonic problem with application we use the variational principle, and depending on the value of the parameter $$a$$ , we obtained uniqueness (non-uniqueness) theorems or present exact formulas for the dimension of the space of solutions.

We consider a reaction-diffusion-advection problem in a perforated medium, with nonlinear reactions in the bulk and at the microscopic boundary, and slow diffusion scaling. The microstructure changes in time; the microstructural evolution is known a priori. The aim of the paper is the rigorous derivation of a homogenized model. We use appropriately scaled function spaces, which allow us to show compactness results, especially regarding the time-derivative and we prove strong two-scale compactness results of Kolmogorov-Simon-type, which allow to pass to the limit in the nonlinear terms. The derived macroscopic model depends on the micro- and the macro-variable, and the evolution of the underlying microstructure is approximated by time- and space-dependent reference elements.

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law $\partial_t \alpha + \mathrm{div}(\boldsymbol{u}f(\alpha)) = 0$ in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity $\mathrm{div}(\boldsymbol{u})$ is of bounded variation instead of the classical assumption that $\mathrm{div}(\boldsymbol{u})$ is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law $\partial_t \alpha + \mathrm{div}(\boldsymbol{F}(t,\boldsymbol{x},\alpha)) = 0$, where $\mathrm{div}_{\boldsymbol{x}}\boldsymbol{F} \not=0$ on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in $L^p$ spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in $\alpha$, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by S. J. Franks et al. [14]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity--pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings., Comment: 35 pages, 7 figures, 13 Tables

The compactness theorem is proved for sequences of functions that have estimates of the higher derivatives in each subdomain of the domain of definition, divided into parts by a sequence of some curves of class W_2^1. At the same time, in the entire domain of determining summable higher derivatives, these sequences do not have. These results allow us to make limit transitions using approximate solutions in problems with an unknown boundary that describe the processes of phase transitions.

In this paper a model of cell proliferation with unequal division is analyzed. The model uses an integral equation to describe the dynamics of the cells density. The analysis of the model is done using tools of the theory developed for the class of perturbed dual semigroups that are solutions of an abstract integral equation. Existence, uniqueness, positivity, compactness and spectral properties of the solution semigroup are derived in order to conclude the asynchronous exponential growth property for the model.

We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly elliptic equations with constant complex coefficients on compact sets in (the particular case of harmonic approximations is not distinguished). The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of a Laurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations). The proof uses an improvement of Vitushkin’s scheme, special geometric constructions, and methods of the theory of singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operators on , the problem considered is technically more difficult than the analogous problem for , 2$?> .

This paper studies the mass-critical variable coefficient nonlinear Schrödinger equation. We first get the existence of the ground state by solving a minimization problem. Then we prove a compactness result by the variational characterization of the ground state solutions. In addition, we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up solutions which are pseudo-conformal transformation of the ground states.

In this article, a weighted problem under boundary Dirichlet condition in the unit ball of $${\mathbb {R}}^{2}$$ is considered. The non-linearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. It is proved that there is a nontrivial positive weak solution to this equation. In the critical case, the compactness condition is not satisfied but a suitable asymptotic condition is used to avoid the non-compactness levels to the energy functional.

We consider a quasistatic contact problem between a viscoelastic material with long-term memory and a foundation. The contact is modelled with a normal compliance condition, a version of Coulomb’s law of dry friction and a bonding field which describes the adhesion effect. We derive a variational formulation of the mechanical problem and, under a smallness assumption, we establish an existence theorem of a weak solution including a regularity result. The proof is based on the time-discretization method, the Banach fixed point theorem and arguments of lower semicontinuity, compactness and monotonicity.

The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.

Global in time weak solutions to the α-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to α-model regularization for the three dimension compressible Euler-Poisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies $$\gamma>{4 \over 3}$$ .

In the present work, we search a new exact solution of Einstein’s field equations using Karmarkar condition of embedded class one space time. The new solution is analyzed graphically as well as analytically to check its viability for compact star modeling. The different parameters of the solution are ascertained by matching the interior space time metric with the Schwarzschild’s exterior space time metric. We tune our solution for the modeling of strange stars EXO1785-248, SAXJ1808.4-3658 and HER X-1 and found their masses and radii as 8.849 km, 1.3 $$M_{\Theta }$$ : 7.951 km, 0.9 $$M_{\Theta }$$ : 8.1 km, 0.85 $$M_{\Theta }$$ , respectively. The physical reliability of the solution depends on the values of the independent parameters b and c. The solution is well behaved for the range of the values $$0.000009\le b\le 0.0039$$ and $$0.0000005\le c\le 0.0000182$$ . Our models satisfy the causality condition and adiabatic index is well behaved everywhere inside the fluid sphere. The compactness parameter is well defined as it does not cross the Buchdahl limit. All the energy conditions hold good inside the compact fluid spheres. The stability of models is assessed via Herrera’s cracking concept. The hydrostatic equilibrium condition is well maintained by our models as gravitational force is counterbalanced by the combined effects of anisotropic force and hydrostatic force. The proper graphical analysis is provided to authenticate the physically admissible character of proposed models.

In this paper, we establish C1,α regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior C1,α regularity result established in Imbert and Silvestre (Adv. Math. 233: 196–206, 2013) for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Holder estimates for equations which are uniformly elliptic when the gradient is either small or large.

We establish a concentration-compactness principle for the Sobolev space W2,p(⋅)(Ω)∩W01,p(⋅)(Ω) that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this r...

We prove a convergence theorem on the moduli space of constant $\sigma_{2}$ metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.

We analyze the properties of the maximal set of initial conditions of the problem of weak practical stability of discrete inclusions with spatial components. We prove compactness and the properties of the boundary and interior of the maximal set of practical stability. In the linear case, we obtain the Minkowski function and the inverse Minkowski function of the optimal set of initial conditions.

In this paper, we first establish the existence of uniform random attractor for 2D stochastic Navier-Stokes equation in H with deterministic non-autonomous external force being normal in L l o c 2 ( R ; V ′ ) , which is the measurable minimal compact set and uniformly attracts bounded random set in H in the sense of pullback. We also show that uniform random attractor with respect to the deterministic non-autonomous functions belonging to some symbol space coincides with uniform random attractor with respect to the initial time. Then we show that the uniform random attractor for the equation under consideration has regularity property in V when deterministic non-autonomous external force being normal in L l o c 2 ( R ; H ) .

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale’s type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

We show that for any given solenoidal initial data in $L^2$ and any solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$, there exist partially regular weak solutions of the Navier-Stokes equations in $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular sets have locally finite $2$-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to overcome the possible lack of compactness arising in the spatially $4$-dimensional setting by using defect measures, which we then incorporate into the partial regularity theory., Various improvements

In this work, we consider sub-critical and critical models for viscoelastic flows driven by pure jump Levy noise. Due to the elastic property, the noise in the equation for the stress tensor is considered in the Marcus canonical form. We investigate existence of a weak martingale solution for stochastic Oldroyd-B models, with full dissipation in whole of R d , d = 2 , 3 . The key ingredients of the proof are classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Pathwise uniqueness, existence of a strong solution and uniqueness in law for the two-dimensional model are also shown. We also prove, in a Poincare domain in two-dimensions, existence of an invariant measure using bw-Feller property of the associated Markov semigroup.

The article examines the efficiency of the structural compact-matrix numerical method in comparison with the classical direct numerical method for calculating the spatial packing density of the simplest hexagonal lattice. The theoretical results of a comparative analysis of the considered numerical meth-ods, as well as a computational experiment, carried out to quantitatively assess their effectiveness, are presented.

We are concerned with global weak solutions to the isentropic compressible Euler equations with cylindrically symmetric rotating structure, in which the origin is included. Due to the presence of the singularity at the origin, only the case excluding the origin $|\vec{x}|\geq1$ has been considered by Chen-Glimm \cite{Chen3}. The convergence and consistency of the approximate solutions are proved by using $L^{\infty}$ compensated compactness framework and vanishing viscosity method. We observe that if the blast wave initially moves outwards and if initial density and velocity decay to zero at certain algebraic rate near the origin, then the density and velocity decay at the same rate for any positive time. In particular, the initial normal velocity is assumed to be non-negative, and there is no restriction on the sign of initial angular velocity.

We study long time dynamics of nonradial solutions to the focusing inhomogeneous nonlinear Schrodinger equation. By using the concentration/compactness and rigidity method, we establish a scatterin...

We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation $$ (-\Delta)^su+V(|z|)u=Q(|z|)u^p\quad \text{in} \ \mathbb{R}^{N},$$ where $N\geq 3,\ 0\frac{N+2s}{N+2s+1}$, and $\theta_1,\ \theta_2>0,\ V_0,\ Q_0>0$. Under various hypotheses on $a_1,\, a_2,\, \alpha,\, \beta$, we establish the existence of infinitely many radial solutions. A key role in our arguments is played by the Lyapunov–Schmidt reduction method.

In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number \begin{document}$ \epsilon $\end{document} for both the compressible system and its differential system with respect to time under uniformly in \begin{document}$ \epsilon $\end{document} small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as \begin{document}$ \epsilon \rightarrow 0 $\end{document} , so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [ 6 ]. Moreover, we also obtain the convergence rates associated with \begin{document}$ L^2 $\end{document} -norm with well-prepared initial data.

In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function.

This manuscript is published in the proceedings of the 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 2021).; International audience; Given a dynamical system $Σp$ with a parameter $p$ taking its values in a fixed interval $\mathcal{Q}$, we present a simple criterion of set inclusion which guarantees that the Euler approximate solutions of $Σp_0$ for some value $p_0$ $∈$ $\mathcal{Q}$ converge to a limit cycle $\mathcal{E}$. Moreover, we characterize a compact set $\mathcal{I}$ containing $\mathcal{E}$ which is invariant for the exact solutions of $Σp$ whatever the value of $p$∈ $\mathcal{Q}$. We illustrate the application of our method on the example of a parametric Van der Pol system driven by a periodic input.

In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.

We prove the existence of a relative finite-energy solution of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p ( ρ ) / ρ = O ( 1 ) in the limit ρ → ∞ . This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier–Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier–Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.

In the present paper, we extend the notion of rough ideal lacunary statistical quasi Cauchy triple difference sequence of order α using the concept of ideals, which automatically extends the earlier notions of rough convergence and rough statistical convergence. We prove several results associated with this notion.

This study presents an optimized gravity-sparse inversion method. The proposed method minimizes the global objective function using interior-point method for boundary constraints and a general weighting function comprising the depth, compactness, and kernel weighting functions of the density models. For the compactness weighting function, practical experiments demonstrate that the recovered model becomes more compact with an increasing value for the relative exponential factor β. However, if no appropriate boundary-constraint method is applied, the inversion results cannot be controlled within the designated constraint bounds when β needs to be set to a large value to obtain compact inversion results. The interior-point method allows the use of a larger β to obtain more compact inversion results without violating the boundary constraints. Additionally, models in close proximity can more clearly be recognized using this method. To improve the computational efficiency and obtain a more accurate regularization parameter, the preconditioned conjugate gradient and L-curve, or line search methods, were also applied. The proposed method was applied for three synthetic examples: two positive bodies adjacent to each other at different depths inverted using noise-free gravity anomaly data, three bodies (positive or negative) at different depths inverted using noise-free or contaminated gravity anomaly data, and three bodies (positive or negative) characterized by a certain dip angle inverted using contaminated gravity anomaly data. This method was also applied for the inversion of a Woodlawn sulfide body, Missouri iron ore body, and granitoid rock body in the Rio Maria region in the state of Para, Brazil. In all six test cases, larger β values were used and the density models were recovered with sharper boundaries within the designated bounds.

We study the unique solvability of the mixed biharmonic problem with the Steklov-type and Farwig conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight $$|x|^{a}$$ . Depending on the value of the parameter $$a$$ , we obtained uniqueness (non-uniqueness) theorems of this problem or present exact formulas for the dimension of the space of solutions.

This paper aims at analyzing the inversion with the mostly used model weighting functions, for both gravity and DC resistivity data. We show that the model weighting function built with depth weighting and compacting factor, formerly formulated for the gravity and magnetics problems, can be useful also for DC resistivity data. We provide a number of synthetic cases to discuss the pros and cons of each model-weighting function. For gravity and DC resistivity data, the comparison was made using the depth weighting with different exponents, the compactness and, for the DC resistivity nonlinear problem, the roughness matrix under the L1- and L2-norm Constrained Optimization. As for the depth weighting, the value of the β exponent is decisive for the gravity problem, ranging from very low values for interfaces to 1 for compact sources. DC resistivity data inversion is less sensitive to β but the above indicated choice leads to a faster convergence. Similarly, the role of compactness is decisive for reconstructing a compact source from gravity, while for DC resistivity it is especially useful to warrant an even faster convergence. Using the roughness matrix tends instead to provide a decrease in resolution at depth. We obtained interesting results for different types of DC resistivity arrays: the weighting function built with depth-weighting and compactness yields a more coherent source reconstruction than that using the roughness matrix. We also analyze two different real DC resistivity cases, which confirm, again, the usefulness of the depth weighting and compactness to model the deep resistive sources.

Consider the Langevin process, described by a vector (position,momentum) in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open bounded and connected set of $\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\mathcal{O}\times\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.