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Abstract: Abstract We investigate compressible micropolar fluids on a time-dependent domain with slip boundary conditions. Our contribution in this paper is threefold. Firstly, we establish the local existence of the strong solution. Secondly, the global existence of weak solutions is shown. The third one is the weak-strong uniqueness principle for slip boundary conditions. There are several new ideas developed by us to overcome the difficulties caused by the coupled terms and slip boundary conditions. PubDate: 2022-05-14

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Abstract: Abstract We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated. PubDate: 2022-05-10

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Abstract: Abstract Recently, trans-S-manifolds have been defined as a wide class of metric f-manifolds which includes, for instance, f-Kenmotsu manifolds, S-manifolds and C-manifolds and generalize well-studied trans-Sasakian manifolds. The definition of trans-S-manifolds is formulated using the covariant derivative of the tensor f and although this formulation coincides with the characterization of trans-Sasakian manifolds in such a particular case, this latter type of manifolds were not initially defined in this way but using the Gray-Hervella classification of almost Hermitian manifolds. The aim of this paper is to study how (almost) trans-S-manifolds relate with the Gray-Hervella classification and to establish both similarities and differences with the trans-Sasakian case. PubDate: 2022-04-30

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Abstract: Abstract In this paper, we consider the following coupled gradient-type quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{*{20}l} - {\text{div}} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &{}{\hbox { in }}\Omega ,\\ - {\text{div}} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left( x, u, v\right) &{}{\hbox { in }}\Omega ,\\ u = v = 0 &{}{\hbox { on }}\partial \Omega , \end{array} \right. \end{aligned}$$ where \(\Omega\) is an open bounded domain in \({\mathbb {R}}^N\) , \(N\ge 2\) . We suppose that some \(\mathcal {C}^{1}\) –Carathéodory functions \(A, B:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) exist such that \(a(x,t,\xi ) = \nabla _{\xi } A(x,t,\xi )\) , \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t} (x,t,\xi )\) , \(b(x,t,\xi ) = \nabla _{\xi } B(x,t,\xi )\) , \(B_t(x,t,\xi ) =\frac{\partial B}{\partial t}(x,t,\xi )\) , and that \(G_u(x, u, v)\) , \(G_v(x, u, v)\) are the partial derivatives of a \(\mathcal {C}^{1}\) –Carathéodory nonlinearity \(G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) . Roughly speaking, we assume that \(A(x,t,\xi )\) grows at least as \((1+ t ^{s_1p_1}) \xi ^{p_1}\) , \(p_1 > 1\) , \(s_1 \ge 0\) , while \(B(x,t,\xi )\) grows as \((1+ t ^{s_2p_2}) \xi ^{p_2}\) , \(p_2 > 1\) , \(s_2 \ge 0\) , and that G(x, u, v) can also have a supercritical growth related to \(s_1\) and \(s_2\) . Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones. PubDate: 2022-04-30

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Abstract: Abstract We derive and subsequently analyze an exact solution of the geophysical fluid dynamics equations which describes equatorial flows (in spherical coordinates) and has a discontinuous fluid stratification that varies with both depth and latitude. More precisely, this solution represents a steady, purely–azimuthal equatorial two-layer flow with an associated free-surface and a discontinuous distribution of the density which gives rise to an interface separating the two fluid regions. While the velocity field and the pressure are given by means of explicit formulas, the shape of the free surface and of the interface are given in implicit form: indeed we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface’s shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. We also provide a regularity result for the interface defining function under certain assumptions on the velocity field. PubDate: 2022-04-28

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Abstract: Abstract The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over projective spaces and certain Blow-up of projective spaces. PubDate: 2022-04-24

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Abstract: Abstract The purpose of this paper is to establish some weighted Sobolev inequalities, which are the borderline cases of the Sobolev embedding on the upper half-space. We use this inequality to derive a weighted Trudinger–Moser-type inequality. Our proofs rely on a simple decomposition which is rearrangement free. As an application of our results, we address the existence of solutions for a class of elliptic problems with exponential critical growth. PubDate: 2022-04-23

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Abstract: Abstract In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space \(\mathbb {E}^3\) is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space \(\mathbb {L}^3\) . In this article, we introduce a new deformation family that continuously connects minimal surfaces in \(\mathbb {E}^3\) and maximal surfaces in \(\mathbb {L}^3\) , and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space \(\mathbb {I}^3\) appear in the middle of the above deformation family. We also prove another type of Krust’s theorem for this family, which implies that the graphness of such minimal surfaces in \(\mathbb {I}^3\) strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory. PubDate: 2022-04-20

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Abstract: Abstract We prove that the first eigenvalue of the fractional Dirichlet–Laplacian of order s on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for \(1/2<s<1\) and we show that this condition is sharp, i.e., for \(0<s\le 1/2\) such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect to s, as it permits to recover a classical result by Makai and Hayman in the limit \(s\nearrow 1\) . The paper is as self-contained as possible. PubDate: 2022-04-15

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Abstract: Abstract A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space \({\mathbb{S}}^n\) of symmetric \(n\times n\) matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in \({\mathbb{S}}^n\) for \(n\le 7\) (resp. \(n\le 5\) ), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in \(\mathbb{S}^n\) for \(n\le 5\) , these obstructions show that our list of degenerations is complete . For \(n=6\) , the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions and then used it to compute the degenerations between Jordan nets in \(\mathbb {S}^7\) and Jordan webs in \(\mathbb {S}^n\) for \(n=4,5\) . PubDate: 2022-04-06

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Abstract: Abstract We examine a variational free boundary problem of Alt–Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior \(C^2\) -regularity of minimizers and that the free boundary consists of finitely many \(C^2\) -hypersurfaces. With the aid of these results, we can prove that minimizers are in general not unique. We investigate radial symmetry of minimizers and compute radial solutions explicitly. PubDate: 2022-04-06

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Abstract: Abstract The concept of slice regular function over the real algebra \(\mathbb {H}\) of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let \(\varOmega \subset \mathbb {H}\) be a domain, i.e., a non-empty connected open subset of \(\mathbb {H}=\mathbb {R}^4\) . Suppose that \(\varOmega\) intersects \(\mathbb {R}\) and is invariant under rotations of \(\mathbb {H}\) around \(\mathbb {R}\) . A function \(f:\varOmega \rightarrow \mathbb {H}\) is slice regular if it is of class \(\mathcal {C}^1\) and, for all complex planes \(\mathbb {C}_I\) spanned by 1 and a quaternionic imaginary unit I ( \(\mathbb {C}_I\) is a ‘complex slice’ of \(\mathbb {H}\) ), the restriction \(f_I\) of f to \(\varOmega _I=\varOmega \cap \mathbb {C}_I\) satisfies the Cauchy–Riemann equations associated with I, i.e., \(\overline{\partial }_If_I=0\) on \(\varOmega _I\) , where \(\overline{\partial }_I=\frac{1}{2}\big (\frac{\partial }{\partial \alpha }+I\frac{\partial }{\partial \beta }\big )\) . Given any positive natural number n, a function \(f:\varOmega \rightarrow \mathbb {H}\) is called slice polyanalytic of order n if it is of class \(\mathcal {C}^n\) and \(\overline{\partial }_I^{\,n}f_I=0\) on \(\varOmega _I\) for all I. We define global slice polyanalytic functions of order n as the functions \(f:\varOmega \rightarrow \mathbb {H}\) , which admit a decomposition of the form \(f(x)=\sum _{h=0}^{n-1}\overline{x}^hf_h(x)\) for some slice regular functions \(f_0,\ldots ,f_{n-1}\) . Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each \(n\ge 2\) , we give examples of slice polyanalytic functions of order n, which are not global. The aim of this ... PubDate: 2022-04-06

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Abstract: Abstract In this paper, we establish a local regularity result for \(W^{2,p}_{{\mathrm {loc}}}\) solutions to complex degenerate nonlinear elliptic equations \(F(D^2_{\mathbb {C}}u)=f\) when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01129-y

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Abstract: Abstract Let \(X^n\) be a nonsingular hypersurface of degree \(d\ge 2\) in the projective space \(\mathbb {P}^{n+1}\) defined over a finite field \(\mathbb {F}_q\) of q elements. We prove a Homma–Kim conjecture on a upper bound about the number of \(\mathbb {F}_q\) -points of \(X^n\) for \(n=3\) , and for any odd integer \(n\ge 5\) and \(d\le q\) . PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01131-4

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Abstract: Abstract In the realm of conformal geometry, we give a parametric classification of the hypersurfaces in Euclidean space that admit nontrivial conformal infinitesimal variations. A parametric classification of the Euclidean hypersurfaces that allow a nontrivial conformal variation was obtained by E. Cartan in 1917. In particular, we show that the class of hypersurfaces studied here is much larger than the one characterized by Cartan. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01136-z

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Abstract: Abstract Given certain functions \(\varphi ,q,h\) and a nonnegative constant \(\lambda \) , we establish regularities for the free boundary problem $$\begin{aligned} {\mathcal {J}}(u)=\int _{\varOmega }(G( \nabla u )+qF(u^+)+hu+\lambda \chi _{\{u>0\}} )\text{d}x\rightarrow \text{min}, \end{aligned}$$ over the set \(\{u\in W^{1,G}(\varOmega ): u-\varphi \in W^{1,G}_{0}(\varOmega )\}\) in the setting of Orlicz spaces, where the functions G and F satisfy the structural conditions of Tolksdorf’s type. Moreover, F allows for subcritical exponents. The main results obtained in this paper include: the local \(C^{1,\alpha }\) - and Log-Lipschitz continuities of minimizers in the subcritical case for \(\lambda =0\) and \(\lambda \ge 0\) , respectively; the growth rates near the free boundary for non-negative minimizers in the subcritical case for \(\lambda \ge 0\) , which give the optimal growth rates of non-negative minimizers for the p-Laplacian problems; the local Lipschitz continuity of non-negative minimizers for \(\lambda >0\) under the natural growth condition that \(F(t)\lesssim 1+G(t)\) for \(t \ge 0\) . All the results presented in this paper are new even for the free boundary problems of p-Laplacian. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01134-1

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Abstract: Abstract Let A(t) \((t\ge 0)\) be an unbounded variable operator on a Banach space \({\mathcal {X}}\) with a constant dense domain, and B(t) be a bounded operator in \({\mathcal {X}}\) . Assuming that the evolution operator U(t, s) \((t\ge s)\) of the equation \(\mathrm{d}x(t)/\mathrm{d}t=A(t)x(t)\) is known we built the evolution operator \(\tilde{U}(t,s)\) of the equation \(\mathrm{d}y(t)/\mathrm{d}t=(A(t)+B(t))y(t)\) . Besides, we obtain C-norm estimates for the difference \(\tilde{U}(t,s)-U(t,s)\) . We also discuss applications of the obtained estimates to stability of the considered equations. Our results can be considered as a generalization of the well-known Dyson–Phillips theorem for operator semigroups. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01139-w

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Abstract: Abstract In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev’s type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01141-2

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Abstract: Abstract Let \({\mathfrak {X}}\) be a group class. A group G is an opponent of \({\mathfrak {X}}\) if it is not an \({\mathfrak {X}}\) -group, but all its proper subgroups belong to \({\mathfrak {X}}\) . Of course, every opponent of \({\mathfrak {X}}\) is a cohopfian group and the aim of this paper is to describe the smallest group class containing \({\mathfrak {X}}\) and admitting no such a kind of cohopfian groups. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01146-x

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Abstract: Abstract Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group \(S_d\) . We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points. PubDate: 2022-04-01 DOI: 10.1007/s10231-021-01132-3