Introduces fractional Sobolev spaces, gives their most important properties and explains their connection with interpolation theory.

Summarises basic information concerning Banach spaces and function spaces.

Derives inequalities of Rellich type in both classical and fractional form.

In this chapter we discuss words and infinite words (sequences) in more detail, giving complete definitions. A small amount of this material is repeated from the previous chapter.

In this chapter we will examine about 80 different fundamental properties of automatic sequences, and show how each one can be encoded by first-order logical formulas. We then use Walnut to re-derive new proofs of known results, or prove new results, concerning some famous automatic sequences. You can use these examples to learn what Walnut is capable of, but also as a ‘catalogue’ of first-order statements of fundamental properties of sequences.

We resort to approximation methods to produce a regularity theory for degenerate models, both in the variational and nonvariational settings. First, we consider a $p$-Poisson equation and approximate it with the Laplace operator. The method allows us to control the distance of the solutions to a harmonic function by a small parameter and a quantity of class $C^1$. This fact unlocks an $C^{1,1-}$ regularity result for the solutions to the $p$-Poisson problem. In the nonvariational context, we consider fully nonlinear equations degenerating as a power of the gradient. Here we detail how approximation methods characterize the optimal regularity of the solutions in terms of the degeneracy rate. A key step is a cancellation lemma in the viscosity sense.

We discuss the role of differentiability in the regularity of the solutions to homogeneous, fully nonlinear elliptic equations. The first part of the chapter concerns flat solutions to equations driven by operators of class $C^2$. Owing to Savin, this result ensures that solutions are of class $C^{2,\alpha}$ and depend only on uniform ellipticity in a neighborhood of the origin in the space of symmetric matrices. We highlight the relevance of differentiability as a condition replacing convexity in the Evans–Krylov theory. The second part of the chapter details the partial regularity result, which prescribes an upper bound for the Hausdorff dimension of the singular set for the viscosity solutions to $F(D^2u)=0$. This result cleverly combines the smoothness of flat solutions and Lin's integral estimates.

In this chapter we give yet another fundamental way to think about automatic sequences, based on first-order logic. This revolutionary approach is originally due to Büuchi, with elaborations and additions by Bruyére, Hansel, Michaux, and Villemaire, and their ideas form the basis for this book. A good reference for the material in this section is the wonderful survey paper by these last four authors [56].

This chapter introduces the basics of elliptic partial differential equations. We detail the notions of viscosity solutions, both in the case of continuous and measurable ingredients, strong solutions, and establish fundamental properties. Those include results on the existence and stability and the Aleksandrov–Bakelman–Pucci estimate. The chapter also details fundamental results in regularity theory, such as the Krylov–Safonov and Evans–Krylov theories, Lin's integral estimates, and Caffarelli's estimates. We introduce approximation methods to establish Hölder continuity of the gradient for fully nonlinear inhomogeneous equations governed by operators with variable coefficients. To finish the chapter, we discuss the so-called counterexamples of Nadirashvili and Vladut, and detail an application. In fact, we prove that a continuous function may satisfy viscosity inequalities driven by the extremal operators but fail to satisfy a uniformly elliptic equation.

We examine the Isaacs equations through approximation methods. Because this equation lacks convexity and differentiability, important developments in regularity theory are not available. We approximate an Isaacs operator by a Bellman one, with fixed coefficients, and transmit regularity from the latter to the former. It allows us to develop a regularity for the solutions to the Isaacs equation in Sobolev and Hölder spaces (including classical solutions).

Walnut is free software originally designed and written in Java by Hamoon Mousavi [278], and recently modified by Aseem Raj Baranwal, Laindon C. Burnett, Kai Hsiang Yang, and Anatoly Zavyalov. This book is based on the most recent version of Walnut, called Walnut 3.7, which is available for free download at