We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair , where is the characteristic function of a set of finite perimeter and is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

]]>We consider the following inhomogeneous problems

where is a smooth and bounded domain in general dimensional space , is a small parameter and function is positive. We respectively obtain the locations of interior transition layers of the solutions of the above transition problems that are -local minimizer and global minimizer of the associated energy functional.
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We prove that, given , the Fourier coefficients of functions in belong to , and that, given , the Fourier series of sequences in belong to . Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every and every , there is a Schauder basis of whose conditionality constants grow as , and there is an almost greedy basis of whose conditionality constants grow as .

]]>We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.

]]>We deal with analytic three-dimensional symmetric systems whose origin is a Hopf-zero singularity. Once it is not completely analytically integrable, we provide criteria on the existence of at least one functionally independent analytic first integral. In the generic case, we characterize the analytic partially integrable systems by using orbitally equivalent normal forms. We also solve the problem through the existence of a class of formal inverse Jacobi multiplier of the system.

]]>This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain with being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field . Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in as . As a consequence, the solution converges to its horizontal average asymptotically.

]]>This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

with positive parameters , . When posed under no-flux boundary conditions in a smoothly bounded domain , and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020, J. Differ. Equ. 268, 4973–4997). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate , the global classical solution is uniformly bounded and exponentially stabilizes to the constant equilibrium in the topology as .
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In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is and the viscosity of the hosting fluid is , . The proof is carried out by the construction of solutions for large and small with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter , as long as is outside the interval , where the positive constants and are the extension constants that depend only on the geometry of the periodic pore space of the material.

]]>In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space . We also prove an existence result of standing waves and prove their orbital stability.

]]>Let be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of with a small enough sphere centred at the origin in . If is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

]]>The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in , , which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the -norm of is exponential decay.

]]>Any Lipschitz map between two pointed metric spaces may be extended in a unique way to a bounded linear operator between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for to be compact in terms of metric conditions on . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that is compact if and only if it is weakly compact.

]]>We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly -stable. As a result, they admit an orbifold Kähler–Einstein metric.

]]>We prove the existence of nontrivial ground state solutions of the critical quasilinear Hénon equation It is a new problem in the sense that the signs of the coefficients of critical terms are opposite.

]]>In this paper, we give the generic classification of the singularities of 3-parameter line congruences in . We also classify the generic singularities of normal and Blaschke (affine) normal congruences.

]]>There is a problem with the proofs of [1], Lemma 4.4 and the related Theorems 4.5, 4.8 and 4.12 regarding the computation of zero-divisor cup-length of real Grassmann manifolds . The correct statements and improved estimates of the topological complexity of will appear in a separate paper by M. Radovanović [2].

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