We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Let $\sigma \in (0,\,2)$, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$, where $\mathbf {0}$ denotes the origin of $\mathbb {R}^n$, and $a$ be a non-negative and bounded measurable function on $\mathbb {R}^n$. In this paper, we obtain the boundedness of the non-local elliptic operator

\[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]

$\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$\lambda \in (0,\,\infty )$

$Lu-\lambda u=f$

$\mathbb {R}^n$

$f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$H^1(\mathbb {R}^n)$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$L^p(\mathbb {R}^n)$

$p\in (1,\,\infty )$

$p=1$

$p=\infty$

In this paper, we discuss the solvability of the p-k-Hessian inequality $\sigma _{k}^{\frac 1k} ( \lambda ( D_{i} (|Du|^{p-2}$ $ D_{j}u ) ) ) \geq f(u)$ on the entire space $\mathbb {R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller–Osserman condition. Furthermore, we obtain the optimal regularity of solution.

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:

• Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:

  1. (1) For all high-enough λ,Khas no long splitting chains.

  2. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.

  3. (3) For all high-enough λ,Khas a unique limit model of cardinality λ.

  4. (4) For all high-enough λ,Khas a superlimit model of cardinality λ.

  5. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.

  6. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.

This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Let $p$ be an odd prime. In this note, we show that a finite group $G$ is solvable if all degrees of irreducible complex characters of $G$ not divisible by $p$ are either 1 or a prime.

We consider a model for flow in a porous medium with a fracture in which the flow in thefracture is governed by the Darcy−Forchheimerlaw while that in the surrounding matrix is governed byDarcy’s law. We give an appropriate mixed, variational formulation and show existence anduniqueness of the solution. To show existence we give an analogous formulation for themodel in which the Darcy−Forchheimerlaw is the governing equation throughout the domain. We showexistence and uniqueness of the solution and show that the solution for the model withDarcy’s law in the matrix is the weak limit of solutions of the model with theDarcy−Forchheimerlaw in theentire domain when the Forchheimer coefficient in the matrix tends toward zero.

This article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence in L∞-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.

This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin (MLPG) method with radial basis function (RBF) kernels generated trial spaces. Local weak-form testings are done with step-functions. It is proved that subject to sufficiently many appropriate testings, solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed. Moreover, an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation. Numerical results (in double precision) give good agreement with the provided theory.

c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

The soil water movement modelgoverned by the initial-boundary value problem for a quasilinear1-D parabolic equation with nonlinear coefficients is considered.The generalized statement of the problem is formulated. Thesolvability of the problem is proved in a certain class offunctional spaces. The data assimilation problem for this model isanalysed. The numerical results are presented.

A class of variationaldata assimilation problems on reconstructingthe initial-value functions is considered for the models governed byquasilinear evolution equations. The optimality system is reduced to theequation for the control function. The properties of the control equation are studied and thesolvability theorems are proved for linear and quasilinear data assimilationproblems. The iterative algorithms for solving the problem are formulated andjustified.