Inhomogeneous-growth problems including a linear-growth term

Calculus of variations is a branch of mathematics, whose main goal is finding minima of functions defined on infinite-dimensional objects. It first appeared in relation to mathematical physics: it was observed by Euler and Lagrange in the XVIII century that behaviour of systems governed by some partial differential equations can often be equivalently described by a minimisation problem. This fact is the basis of the use of variational methods to problems in physics. Then, the infinite- dimensional objects are function spaces, and the minimised object is a functional, whose input is a function and its output is a number. A typical example is the heat flow, when the corresponding functional is the integral of the square of the gradient of a function. In every problem at the interface of the calculus of variations and partial differential equations, three issues are most relevant. The first one is existence of solutions in some properly chosen class. The second one is uniqueness of solutions in said class, and the third one is their regularity; this term covers any additional properties of the solution such as smoothness, energy bounds, or boundary behaviour. This applies both to elliptic problems (modelling stationary states) and parabolic problems (modelling evolution in time). It turns out that many properties of solutions depend on the rate of growth of the minimised functional. For instance, linear equations correspond to quadratic growth of the functional, and in this case one can obtain very strong results. Most of the modern techniques are best suited to the case when the growth is given by some power greater than 1. Let us highlight two cases outside of this framework: when the functional has linear growth and when the growth is inhomogeneous, i.e. its rate may depend on location or direction. In this project, we study problems which are at the interface between these two cases: the main goal of the project is to study parabolic and elliptic problems featuring both inhomogeneous growth and linear growth. To be exact, we are interested in equations in which associated functionals include a term with linear growth and a term with faster growth. They appear naturally in models of crystal growth and Bingham fluids. Moreover, they are used in some algorithms in image processing. So far, there have been very few rigid results concerning such problems. We plan to study existence, uniqueness, and regularity of solutions, and to complement it with an analysis of qualitative properties of solutions such as formation of facets or approximate behaviour of solutions to evolution problems for large times. The underlying theme is that there is a competition between the linear term and the superlinear term: the former favors phase transitions and formation of discontinuities, while the latter may have regularising properties.

The project is in the area of calculus of variations, which is a branch of mathematics whose main goal is finding minima of functions defined on infinite-dimensional objects. It is strongly related to partial differential equations through a formalism called Euler-Lagrange equations. This allows for the use of variational methods in problems in (among others) physics, geometry, and image processing. The main difficulty in the project stems from the following fact: the studied functionals (in this context, objects whose input is a function and the output is a number) do not satisfy the classical conditions on the growth of the functional. Most of the modern techniques are best suited to the case when the growth is given by some power greater than 1, but the main focus of the project is on the case when the functional exhibits both inhomogeneous growth (i.e., its rate may depend on location or direction) and linear growth. Thus, at the start of the project very few results were available; this applies to both elliptic problems and parabolic problems, which model stationary states and evolution in time respectively. Therefore, the first line of results concerns well-posedness for several types of variational problems and evolution equations, including existence and uniqueness of solutions. While many of the results are stated in a more general setting, our model problems are of the following three types. In the first one, the functional exhibits linear growth in some region of the ambient space and faster growth away from this region; our main examples are the double-phase and variable-exponent models. The second one is that the functional exhibits different rates of growth in different directions, with linear growth in at least one direction, leading to the so-called anisotropic p-Laplacian. The third type are problems on discrete structures such as graphs or networks, with the anisotropic growth embedded into the description of the structure. A second type of results achieved in this project concerns the qualitative behaviour of solutions. For the problems outlined in the previous paragraph, we primarily studied their regularity and asymptotic behaviour. In other words, we verified if the solution has some additional properties such as smoothness or local energy bounds, and described its approximate behaviour for large times (for evolution problems). Furthermore, we proved several results concerning characterisation of solutions, allowing us to provide a set of verifiable conditions which uniquely describe a solution. From this we obtained more information on the structure of solutions, in particular in discrete settings; most importantly, in the case of a double-phase energy, we used this characterisation to develop a new denosing model in image processing which simultanously preserves the existing edges within the image and avoids the so-called staircasing effect.