In this talk, we introduce a parametric bilinear fractional integral operator and obtain uniform bounds via Marcinkiewicz interpolation. This operator is motivated by a reformulation of Euler-Riesz systems, where the nonlocal term in the momentum equation is expressed as the divergence of a tensorial integral operator. This reformulation is relevant as it leads to an a priori higher integrability estimate for the density of finite-energy solutions. The uniform bounds obtained for the parametric fractional integrals then yield bounds for the tensorial integral operator. This is based on joint work with L. Grafakos and A. E. Tzavaras.

In this talk we consider differential operators of the form $\text{div}(A \, \text{grad})$ where the coefficients of $A$ are measurable and bounded functions and the eigenvalues of $A$ are positive and uniformly bounded from $0$. A solution on a domain $D \subset \mathbb{R}^d$ is a function that satisfies\[ \operatorname{div}(A \, \operatorname{grad}) u = 0 \text{ on } D. \]

We examine boundary behavior of the gradient of solutions to such differential operators. The classical case, where the differential operator in question is the Laplacian, is well understood for $C^2$ domains due to Havin and Mozolyako [1] and later also for Lipschitz domains due to Müller and Riegler [2]. This talk mentions techniques to obtain estimates on the radial variation of positve harmonic functions and also gives an outlook of how to adapt the existing techniques to general uniformly elliptic divergence-form operators or solutions on more complicated domains.

1. Pavel A. Mozolyako and Viktor P. Khavin. Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra i Analiz 28.3 (2016), pp. 67–110. issn: 0234-0852. doi: 10.1090/spmj/1454. url: https://doi.org/10.1090/spmj/1454.
2. Paul F. X. Müller and Katharina Riegler. Radial variation of Bloch functions on the unit ball of Rd. Ark. Mat. 58.1 (2020), pp. 161–178. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2020.v58.n1.a10. url: https://doi.org/10.4310/ARKIV.2020.v58.n1.a10.

What we are calling Emden-Fowler-Hénon (EFH) type equations are elliptic partial differential equations that have the main structure as $-\Delta u=h(x)|u|^{p-1}u$, usually considering $h(x)$ a radial function and with $p$ a subcritical exponent in the sense of Sobolev embeddings, that is, $1 < p < 2^*-1$. Taking $h(x) \equiv 1$, the resulting equation is called Lane-Emden equation which is used in astrophysics to model stellar structures. If we consider $h(x) = |x|^\alpha$, we obtain the so called Hénon equation, proposed in [1] as a perturbation of Lane-Emden equation to study the stability of spherical stellar systems. In mathematics, the Hénon equation plays many important roles in the theory of qualitative analysis of PDE, for instance, by providing a nontrivial example to theorems and working as a guide to develop new tools to fortify the theory. Significant developments have been made by [2, 3, 4, 5] to understand the qualitative properties of the Hénon equation and to set a process to analyze EFH type equations. In this setting, we propose an EFH type equation considering $h(x) = (4|x|(1 − |x|))^\alpha$, which is a weight that presents a behavior unexplored in literature and, we believe, necessary to comprehend the full picture of EFH type equations. We are also interested in understanding how well the developed technics apply to this new configuration.

1. Hénon, M. (1974). Numerical Experiments on the Stability of Spherical Stellar Systems. Symposium - International Astronomical Union, 62, 259-259. https://doi.org/10.1017/s0074180900070662
2. Smets, D., Willem, M. & Su, J. Non-radial ground states for the Hénon equation. Communications In Contemporary Mathematics. 4, 467-480 (2002,8)
3. Cao, D., Peng, S. & Yan, S. Asymptotic behaviour of ground state solutions for the Hénon equation. IMA Journal Of Applied Mathematics. 74, 468-480 (2008,12)
4. Silva, W. & Santos, E. Asymptotic profile and Morse index of the radial solutions of the Hénon equation. Journal Of Differential Equations. 287 pp. 212-235 (2021,6)
5. Mercuri, C. & Santos, E. Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations. Nonlinearity. 32, 4445-4464 (2019,10).

In this talk, we investigate the existence of infinitely many non-constant weak solutions to the quasilinear elliptic equation
\[ -\operatorname{div}(A(x,u)∇ u) + \frac{1}{2} D_u A(x,u)∇ u \cdot ∇ u = g(x,u) -λ u, \quad u∈ H^1(Ω), \] for every \( λ ∈ \mathbb{R} \), where \(Ω \) is a bounded domain with a Lipschitz continuous boundary.

Our approach is variational; however, the associated energy functional is differentiable only along directions \( v ∈ H^1(Ω) ∩ L^∞(Ω) \).

We discuss the notion of weak slope and the critical point theory for continuous functionals, and we apply these concepts to our problem to establish the existence of multiple solutions.

Visualization of mountainous topography is commonly done using a shaded relief algorithm that combines an imaginary light source with a Lambertian reflectance model. The resulting images are intuitive to understand, but they also suffer certain drawbacks. Among these is anisotropy, or directional dependence, whereby the choice of lighting direction favors certain terrain features at the expense of others. Another common issue is lack of visual hierarchy, since small terrain features tend to clutter the image and obscure larger landforms that are actually more prominent. Most attempts to mitigate these problems have stayed within the general paradigm of illumination models.

A newer algorithm called texture shading attempts to address the shortcomings of conventional shaded relief by taking a different approach, not derived from a lighting model. The underlying mathematics uses a fractional Laplacian operator adapted to work on discrete gridded data (in this case, a digital elevation model) by means of a discrete cosine transform. The algorithm is isotropic, and scale-invariant in a particular sense, which gives a clear visual hierarchy to the images, and it highlights the network of ridges and canyons in the terrain. Fractional derivatives appear to be well-suited to terrain data due to the fractal nature of many topographic structures over a wide dynamic range.

The presentation will cover the motivation and goals of the algorithm, the implementation and mathematical details, example results, potential applications, and future work.