Rethinking the Methodological Foundation of Historical Political Science

The basis of a methodology determines whether a research method can fit the core characteristics of a particular academic tradition, and thus, it is crucial to explore this foundation. Keeping in mind the controversy and progress of the philosophy of social sciences, this paper aims to elaborate on four aspects including the cognitive model, the view of causality, research methods, and analysis techniques, and to establish a more solid methodological basis for historical political science. With respect to the “upstream knowledge” of methodology, both positivism and critical realism underestimate the tremendous difference between the natural world and the social world. This leads to inherent flaws in controlled comparison and causal mechanism analysis. Given the constructiveness of social categories and the complexity of historical circumstances, the cognitive model of constructivism makes it more suitable for researchers to engage in macro-political and social analysis. From the perspective of constructivism, the causality in “storytelling,” i.e., the traditional narrative analysis, is placed as the basis of the regularity theory of causality in this paper, thus forming the historical–causal narrative. The historical–causal narrative focuses on how a research object is shaped and self-shaped in the ontological historical process, and thus ideally suits the disciplinary characteristics of historical political science. Researchers can complete theoretical dialogues, test hypotheses, and further explore the law of causality in logic and evidence, thereby achieving the purpose of “learning from history” in historical political science.

Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

Concentrating standing waves for Davey–Stewartson systems

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.