In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n⩽9. This result, that was only known to be true for n⩽4, is optimal: log(1/|x|^2) is a W^{1,2} singular stable solution for n⩾10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n⩽9, stable solutions are bounded in terms only of their L^1 norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are W^{1,2} in every dimension and they are smooth in dimension n⩽9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.

A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

A novel discontinuous Galerkin (DG) method is developed to solve timedependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are L^2 stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal L^2 error estimate of O(h^{k+1}) for polynomials of degree k for semi-discrete DG schemes, and the L2 error of O(h^{k+1} + (\Delta t)^2) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.

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