The purpose of this paper is to study the stability of some unilateral free-discontinuity problems in two-dimensional domains, with the density of the volume part having p-growth, with 1 < p < ∞, under perturbations of the discontinuity sets in the Hausdorff metric.

We prove monotonicity properties of positive weak solutions to – Δpu = f(u) in half-spaces for the case in which (2N + 2)/(N + 2) < p < 2 when zero Dirichlet boundary conditions are imposed. The model nonlinearity is given by f(s) := sq − λsm with q > m ≥ p − 1 and λ > 0.

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.

Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).

In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters ai for which the polynomial xn + a1xn−1 + … + an is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a1 = 0, a2 ≥ −1}.

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

When a liquid fills the semi-infinite space between two concentric cylinders which rotate at different steady speeds, how about the shape of the free surface on top of the fluid? The different fluids will lead to a different shape. For the Newtonian fluid, the meniscus descends due to the centrifugal forces. However, for the certain non-Newtonian fluid, the meniscus climbs the internal cylinder. We want to explain the above phenomenon by a rigorous mathematical analysis theory. In the present paper, as the first step, we focus on the Newtonian fluid. This is a steady free boundary problem. We aim to establish the well-posedness of this problem. Furthermore, we prove the convergence of the formal perturbation series obtained by Joseph and Fosdick in Arch. Ration. Mech. Anal. 49 (1973), 321–380.

For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

In this paper the minimum fundamental gap of a kind of sub-elliptic operator is concerned, we deal with the existence and uniqueness of weak solution for that. We verify that the minimization fundamental gap problem can be achieved by some function, and characterize the optimal function by adopting the differential of eigenvalues.

We develop a theory of minimal models for algebras over a Koszul operad with trivial differential defined over a commutative ring (containing $\mathbb {Q}$ in the symmetric case), not necessarily a field, extending and supplementing the work of Sagave for the associative case. Our minimal models are bigraded and contain a projective resolution of the homology.

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

We consider the global existence for the following fully parabolic chemotaxis system with two populations

\[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]

$\Omega =\mathbb {R}^2$

$\Omega =B_R(0)\subset \mathbb {R}^2$

$\kappa _i,\chi _i>0,$

$i=1,2$

$0=\Delta v-v+u_1+u_2$

$0=\Delta v+u_1+u_2$

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns. 21 (1996), 363–394); Grenier (Commun. Pure Appl. Math. 50 (1997), 821–865); Ukai (J. Math. Kyoto Univ. 26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity 24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

The rigorous derivation of linear elasticity from finite elasticity by means of $\Gamma$-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we compute its linearization for small pressure by $\Gamma$-convergence. We allow for a weakly coercive elastic energy density and we prove strong convergence of minimizers.

We consider the boundary Hardy–Hénon equation

\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]

$B_1(0)\subset \mathbb {R}^{N}$

$(N\geq 3)$

$1$

$0$

$p>0$

$\alpha \in \mathbb {R}$

$0< p<({N+2})/({N-2})$

$\alpha \leq -2$

$p>1$

$\alpha >-2$

$1< p<({N+2})/({N-2})$

$0< p\leq 1$

$\alpha \leq -2$

$0< p<1$

$\alpha >-2$

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

We note that Gabber's rigidity theorem for the algebraic K-theory of henselian pairs also holds true for hermitian K-theory with respect to arbitrary form parameters.