Mathematical interest: 

I do research in Partial Differential Equations (PDEs). I am particularly interested in nonlinear PDEs, free boundary problems,  variational problems, non-local or fractional problems and nonlinear eigenvalue problems.

PhD/Master/Bachelor: If you are a student interested in a PhD, master or bachelor project, I encourage you to contact me to discuss possible projects.

Seminars and events:

Research program: Geometric Aspects of Nonlinear Partial Differential Equations, at Institut Mittag-Leffler

The PDEs and Applications seminar in Uppsala

The KTH-Uppsala PDE days (not active for the moment)

Current teaching: 

Calculus of Variations, Period 3-4, VT2024

Previous teaching: 

Flervariabelanalys SF1626, Period 4, VT2023

Envariabelanalys SF1625, Period 3, VT2023

Topics in PDEs (PhD course, Period 3-4)

Envariabelanalys (Period 1-2, 2021-2022)

Envariabelanalys (Period 1-2, 2020-2021)

Flervariabelanalys (Period 3, 2020)

Envariabelanalys (Period 1-2, 2019-2020)

Flervariabelanalys (Period 4, 2019)

Variationskalkyl (Period 4, 2019)

Funktionslära för ingenjörer (Period 2, 2018)

Transformmetoder (Period 2, 2018)

Scientific events:

Geometric aspects of nonlinear partial differential equations, at Institut Mittag-Leffler, Fall 2022

Geometric, variational and harmonic analytic methods in PDEs,  June 1-5, 2020, Uppsala, cancelled

Swedish Summer PDEs, August 26-28 2019, KTH, Stockholm

Chinese-Swedish workshop in nonlinear PDEs, 3-4 September 2018, KTH, Stockholm ,

KTH-Jiao Tong workshop, March 12, 2018

KTH, Stockholm , Mini-conference in PDEs, December 11-13 2017, KTH, Stockholm 

14th International Conference on Free Boundary Problems: Theory and Applications, Shanghai, July 2017

Meeting of the Catalan, Spanish, Swedish Math Societies, Umeå, June 2017 

Symposium in nonlinear PDEs, June 3-5 2013, Trondheim

Norwegian-Italian Workshop in PDEs, December 2012

Scientific publications:

A free boundary problem with constant Bernoulli-type boundary condition, joint work with Yannick Privat (Nancy), Nonlinear Analysis: Theory, Methods & Applications, Volume 67, Issue 8, 15 October 2007, Pages 2497-2505.

Regularity of the free boundary for a semilinear elliptic problem in two dimensions, joint work with Arshak Petrosyan (Purdue), published in Indiana Univ. Math J. 57 (2008), 3397-3418.

The N-membranes problem, joint work with Abdolrahman Razani (Imam Khomeini International University), published in Bulletin of the IMS, Volume 35, No. 1, April 2009..

The two-phase obstacle problem for the p-laplacian when p~2, joint work with Anders Edquist (KTH), published in Calc. Var. Partial Differential Equations 35 (2009), no. 4, 421–433..

On the two-phase membrane problem with coefficients below the Lipschitz threshold, joint work with Anders Edquist and Henrik Shahgholian (both at KTH), published in Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 6, 2359–2372.

On the penalized obstacle problem in the unit half ball, published in Electron. J. Differential Equations 2010, No. 09, 12 pp..

Regularity of a parabolic free boundary problem with Hölder continuous coefficients, joint work with Anders Edquist (KTH), published in Comm. Partial Differential Equations 37 (2012), no. 7, 1161–1185.

The Hölder infinite Laplacian and Hölder extensions, joint work with Antonin Chambolle (CMAP) and Régis Monneau (CERMICS), published in ESAIM Control Optim. Calc. Var. 18 (2012), no. 3, 799–835.

Optimal regularity of a parabolic free boundary problem of two-phase type with coefficients worse than Lipschitz, , joint work with Jyotshana V. Prajapat (Petroleum Institute), published, Potential Anal. 37 (2012), no. 2, 103–123.

Stability for the Infinity-Laplace Equation with variable exponent, arxiv preprint, joint work with Peter Lindqvist (NTNU), Differential and Integral Equations 25 (2012), no. 5-6, 589-600.

Optimal regularity for the no-sign obstacle problem, arxiv preprint, joint work with John Andersson (Warwick) and Henrik Shahgholian (KTH), Comm. Pure Appl. Math. 66 (2013), no. 2, 245–262.

On the regularity of solutions of the inhomogeneous infinity Laplace equation, arxiv preprint, Proc. Amer. Math. Soc. 142 (2014), no. 1, 277–288., .

Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients , arxiv preprint, Indiana Univ. Math. J. 62 (2013), no. 1, 171–199, joint work with Régis Monneau (CERMICS).

Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions , preprint, Ark. Mat. 52 (2014), no. 1, 21–42, joint work with Mahmoudreza Bazarganzadeh (Uppsala University).

Fractional eigenvalues, arxiv preprint, joint work with Peter Lindqvist (NTNU), Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 795–826.

Optimal Regularity for the parabolic No-Sign Obstacle Problem, arxiv preprint, joint work with John Andersson (Warwick) and Henrik Shahgholian (KTH), published in Interfaces Free Bound. 15 (2013), no. 4, 477–499.

The two-phase fractional obstacle problem, arxiv, SIAM J. Math. Anal. 47 (2015), no. 3, 1879–1905, joint work with Mark Allen and Arshak Petrosyan (both at Purdue University)

Pointwise regularity of the free boundary for the parabolic obstacle problem, arxiv, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 299–347 (online version), joint work with Régis Monneau (CERMICS)

The fractional Cheeger problem, arxiv, Interfaces Free Bound. 16 (2014), 419-458, joint work with Lorenzo Brasco and Enea Parini (both at Aix-Marseille Université)

Regularity of the p-Poisson equation in the plane, arxiv, Journal d'Analyse Matématique 132 (2017), no. 1, 217–228, joint work with Peter Lindqvist (NTNU)

Optimal regularity for the obstacle problem for the p-Laplacian, arxiv, J. Differential Equations 259 (2015), no. 6, 2167–2179 (online version), joint work with John Andersson och Henrik Shahgholian (both at KTH)

Inverse iteration for p-ground states, Proc. Amer. Math. Soc. 144 (2016), no. 5, 2121–2131, arxiv, joint work with Ryan Hynd (UPenn)

A doubly nonlinear evolution for the optimal Poincaré inequality, arxiv, Calc. Var. Partial Differential Equations 55 (2016), no. 4, 55:100, joint work with Ryan Hynd (UPenn)

Hölder estimates for viscosity solutions of equations of fractional p-Laplace type, arxiv, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 5, 23:55

Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, arxiv, Advances in Mathematics Volume 304, 2 January 2017, Pages 300–354, joint work with Lorenzo Brasco (Aix-Marseille Université)

Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution, Anal. PDE 9 (2016), no. 6, 1447–1482., arxiv, joint work with Ryan Hynd (UPenn)

Approximation of the least Rayleigh quotient for degree p homogeneous functionals, arxiv, J. Funct. Anal. 272 (2017), no. 12, 4873–4918., joint work with Ryan Hynd (UPenn)

Perron's Method and Wiener's Theorem for a Nonlocal Equation, arxiv, Potential Anal. 46 (2017), no. 4, 705–737., joint work with Peter Lindqvist (NTNU)

Equivalence of solutions to fractional p-Laplace type equations, arxiv,  J. Math. Pures Appl. (9) 132 (2019), 1–26., joint work with Janne Korvenpää and Tuomo Kuusi (both Aalto University)

Extremal functions for Morrey's inequality in convex domains , arxiv, Math. Ann. 375 (2019), no. 3-4, 1721–1743., joint work with Ryan Hynd (UPenn)

Large time behavior of solutions of Trudinger's equation , arxiv, joint work with Ryan Hynd (UPenn), JDE 274, 2021, Pages 188-230.

Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case,  Adv. Math. 338 (2018), 782–846., joint work with Armin Schikorra and Lorenzo Brasco

Infinity-Harmonic Potentials and Their Streamlines ,  Discrete Contin. Dyn. Syst. 39 (2019), no. 8, 4731–4746.,  joint work with Peter Lindqvist

Lipschitz regularity for a homogeneous doubly nonlinear PDE, SIAM J. Math. Anal. 51 (2019), no. 4, 3606–3624. , joint work with Ryan Hynd

On a comparison principle for Trudinger's equation, joint work with Peter Lindqvist, Advances in Calculus of Variations, https://doi.org/10.1515/acv-2019-0095

Continuity of solutions to a nonlinear fractional diffusion equation, joint work with Lorenzo Brasco and Martin Strömqvist,  J. Evol. Equ. 21 (2021), no. 4, 4319–4381.

A mean value formula for the variational p-Laplacian, joint work with Félix del Teso,  NoDEA Nonlinear Differential Equations Appl. 28 (2021), no. 3, Paper No. 27, 33 pp. 

The Gradient Flow of Infinity-Harmonic Potentials, joint work with Peter Lindqvist,  Adv. Math. 378 (2021), Paper No. 107526, 24 pp.

On ∞-Ground States in the Plane,  joint work with Peter Lindqvist, accepted in Mathematical Research Letters

A finite difference method for the variational p-Laplacian, joint work with Félix del Teso, J. Sci. Comput. 90 (2022), no. 1, Paper No. 67, 31 pp.

Uniqueness of extremals for some sharp Poincaré-Sobolev constants, joint work with Lorenzo Brasco, accepted in Trans. AMS

Large time behavior for a nonlocal nonlinear gradient flow, joint work with Feng Li, accepted in Discrete and Continuous Dynamical Systems

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations, joint work with Prashanta Garain, accepted in  Calc. Var. Partial Differential Equations

Finite difference schemes for the parabolic p-Laplace equation, joint work with Félix del Teso, SeMA Journal. Online first. (2022)

The Infinity-Laplacian in Smooth Convex Domains and in a Square, joint work with Karl K. Brustad and Peter Lindqvist

Decay of extremals of Morrey's inequality, joint work with Ryan Hynd and Simon Larson, accepted in Arkiv för Matematik

Higher Hölder regularity for the fractional p-Laplace equation in the subquadratic case, joint work with Prashanta Garain

On a Hardy-Morrey inequality, joint work with Ryan Hynd and Simon Larson

Higher Hölder regularity for a subquadratic nonlocal parabolic equation, joint work with Prashanta Garain and Alireza Tavakoli

Notes:

The $\infty$-harmonic potential is not always an $\infty$-eigenfunction, arxiv.