Monatshefte für Mathematik最新文献

In this article, we study the uniqueness and stability of nonnegative solutions for a class of semilinear elliptic problems in a ball, when the nonlinearity has more than one zero, negative at the origin and concave.

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class (M_{rho , Lambda }^m({mathbb {T}}times {mathbb {Z}})) (associated to a suitable weight function (Lambda ) on ({mathbb {Z}})) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of M-elliptic pseudo-differential operators on ({mathbb {T}}). Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class (M_{rho , Lambda }^0({mathbb {T}}times {mathbb {Z}})) and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on (L^2({mathbb {T}})). Finally, we provide Gårding’s and Sharp Gårding’s inequality for M-elliptic operators on ({mathbb {Z}}) and ({mathbb {T}}), respectively, and present an application in the context of strong solution of the pseudo-differential equation (T_{sigma } u=f) in (L^{2}left( {mathbb {T}}right) ).

This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval (I_0=left[ 0,frac{1}{2}right] ) or (I_0=left[ -frac{1}{2},frac{1}{2}right] ). We prove asymptotic formulas (lambda (T^{-n}I) =mu (I)(lambda ( I_0) +O(q^n))) for such transformations T, where (lambda ) is the Lebesgue measure on ({mathbb {R}}), (mu ) the normalized T-invariant Lebesgue absolutely continuous measure, I subinterval in (I_0), and (q=0.288) is smaller than the Wirsing constant (q_Wapprox 0.3036).

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces (B^{s}_{p,r}({mathbb {R}})) with ( 1le p,rle infty ) and (s> max{1+frac{1}{p},frac{3}{2}}), which improve the previous work in Sobolev spaces ( H^{s}({mathbb {R}})= B^{s}_{2,2}({mathbb {R}})) with ( s>frac{3}{2}) (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces (B^{s}_{p,r}({mathbb {R}})) with ( 1le ple infty ),( 1le r< infty ) and (s> max{1+frac{1}{p},frac{3}{2}}).

In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily (K_infty )-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.

We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product (mathbb {R}^nrtimes K), where K is a compact subgroup of automorphisms of (mathbb {R}^n). We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.

In this paper, we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here, we begin the study of the iterations of the functions of this class showing briefly its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by (HH^{2}(mathbb {D})) of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form (chi +overline{pi }) where (chi ,pi ) are analytic functions from (mathbb {D}) into (mathbb {D}). We also analyze the space of bounded linear operators on (HH^{2}(mathbb {D})).

In this paper, we are concerned with determining the explicit solution of a Sea-Breeze flow model by selecting a special viscosity function. Firstly, we examine the exact solution when the viscosity function is related to a nonnegative constant coefficient. Further, by employing suitable transformations and forcing terms, we transform the original second order differential equation corresponding to the Sea-Breeze flow model into the Bessel equation and derive the corresponding exact solution. Finally, we determine the exact solution when the viscosity function is related to a nonnegative quadratic function.

Abstract

This article deals with the existence of multiple positive solutions to the following system of nonlinear equations involving Pucci’s extremal operators: $$begin{aligned} left{ begin{aligned} -mathcal {M}_{lambda _1,Lambda _1}^+(D^2u_1)&=f_1(u_1,u_2,dots ,u_n)~~~{} & {} textrm{in}~~Omega , -mathcal {M}_{lambda _2,Lambda _2}^+(D^2u_2)&=f_2(u_1,u_2,dots ,u_n)~~~{} & {} textrm{in}~~Omega , ~~~~~~~~vdots&=~~~~~~~~~~~~ vdots -mathcal {M}_{lambda _n,Lambda _n}^+(D^2u_n)&=f_n(u_1,u_2,dots ,u_n)~~~{} & {} textrm{in}~~Omega , u_1=u_2=dots =u_n&=0~~{} & {} textrm{on}~~partial Omega , end{aligned} right. end{aligned}$$ where (Omega ) is a smooth and bounded domain in (mathbb {R}^N) and (f_i:[0,infty )times [0,infty )dots times [0,infty )rightarrow [0,infty )) are (C^{alpha }) functions for (i=1,2,dots ,n) . The multiplicity result in this work is motivated by the work Amann (SIAM Rev 18(4):620–709, 1976), and Shivaji (Nonlinear analysis and applications (Arlington, Tex., 1986), Dekker, New York, 1987), where the three solutions theorem (multiplicity) has been proved for linear equations. Later on, it was extended for a system of equations involving the Laplace operator by Shivaji and Ali (Differ Integr Equ 19(6):669–680, 2006). Thus, the results here can be considered as a nonlinear analog of the results mentioned above. We also have applied the above results to show the existence of three positive solutions to a system of nonlinear elliptic equations having combined sublinear growth by explicitly constructing two ordered pairs of sub and supersolutions.

Abstract

For two real bases (q_0, q_1 > 1) , we consider expansions of real numbers of the form (sum _{k=1}^{infty } i_k/(q_{i_1}q_{i_2}ldots q_{i_k})) with (i_k in {0,1}) , which we call ((q_0,q_1)) -expansions. A sequence ((i_k)) is called a unique ((q_0,q_1)) -expansion if all other sequences have different values as ((q_0,q_1)) -expansions, and the set of unique ((q_0,q_1)) -expansions is denoted by (U_{q_0,q_1}) . In the special case (q_0 = q_1 = q) , the set (U_{q,q}) is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases ((q_0, q_1)) with trivial (U_{q_0,q_1}) from those with non-trivial (U_{q_0,q_1}) is the graph of a function (mathcal {G}(q_0)) that we call generalized golden ratio. Similarly, the curve separating pairs ((q_0, q_1)) with countable (U_{q_0,q_1}) from those with uncountable (U_{q_0,q_1}) is the graph of a function (mathcal {K}(q_0)) that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in (q_0) and (q_1) , that (mathcal {G}) and (mathcal {K}) are continuous, strictly decreasing, hence almost everywhere differentiable on ((1,infty )) , and that the Hausdorff dimension of the set of (q_0) satisfying (mathcal {G}(q_0)=mathcal {K}(q_0)) is zero. We give formulas for (mathcal {G}(q_0)) and (mathcal {K}(q_0)) for all (q_0 > 1) , using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems.