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On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity

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Abstract

We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem

$$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$

where \(1 \le p<q<2\), \(\lambda _+ >0\), \(\lambda _- \ge 0\). The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.

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References

  1. Aldushin, A.P., Khaikin, B.I.: Combustion of mixtures forming condensed reaction products. Combustion, Explosion and Shock Waves. 10, 1573–8345, 1974

    Article  Google Scholar 

  2. Alt, H.W., Phillips, D.: A free boundary problem for semilinear elliptic equations. J. Reine Angew. Math. 368, 63–107, 1986

    MathSciNet  Google Scholar 

  3. Andersson, J., Shahgholian, H., Weiss, G.S.: Uniform regularity close to cross singularities in an unstable free boundary problem. Comm. Math. Phys. 296(1), 251–270, 2010

    Article  ADS  MathSciNet  Google Scholar 

  4. Andersson, J., Shahgholian, H., Weiss, G.S.: On the singularities of a free boundary through Fourier expansion. Invent. Math. 187(3), 535–587, 2012

    Article  ADS  MathSciNet  Google Scholar 

  5. Andersson, J., Weiss, G.S.: Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem. J. Differ. Equ. 228(2), 633–640, 2006

    Article  ADS  MathSciNet  Google Scholar 

  6. Aronszajn, N., Krzywicki, A., Szarski, J.: A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat. 4(417–453), 1962, 1962

    MathSciNet  Google Scholar 

  7. Arya, V., Banerjee, A.: Strong backward uniqueness for sublinear parabolic equations. NoDEA, Nonlinear Differ. Equ. Appl. 27(6), 17, 2020 Id/No 50.

    Article  MathSciNet  Google Scholar 

  8. Banerjee, A., Garofalo, N., Manna, R.: Carleman estimates for baouendi-grushin operators with applications to quantitative uniqueness and strong unique continuation. Appl. Anal., (online first) 2020.

  9. Banerjee, A., Manna, R.: Space like strong unique continuation for sublinear parabolic equations. J. Lond . Math. Soc., II. Ser. 102(1), 205–228, 2020

    Article  MathSciNet  Google Scholar 

  10. Beck, J.M., Volpert, V.A.: Nonlinear dynamics in a simple model of solid flame microstructure. Physica D 182(1), 86–182, 2003

    Article  ADS  MathSciNet  Google Scholar 

  11. Bers, L.: Local behavior of solutions of general linear elliptic equations. Comm. Pure Appl. Math. 8, 473–496, 1955

    Article  MathSciNet  Google Scholar 

  12. Blank, I.: Eliminating mixed asymptotics in obstacle type free boundary problems. Commun. Partial Differ. Equ. 29(7–8), 1167–1186, 2004

    Article  MathSciNet  Google Scholar 

  13. Caffarelli, L.A., Friedman, A.: Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations. J. Differ. Equ. 60, 420–433, 1985

    Article  ADS  MathSciNet  Google Scholar 

  14. Chanillo, S., Kenig, C.E.: Weak uniqueness and partial regularity for the composite membrane problem. J. Eur. Math. Soc. 10(3), 705–737, 2008

    Article  MathSciNet  Google Scholar 

  15. Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. Contemp. Math. 268, 61–81, 1999

    Article  MathSciNet  Google Scholar 

  16. Chanillo, S., Grieser, D., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Comm. Math. Phys. 214(2), 315–337, 2000

    Article  ADS  MathSciNet  Google Scholar 

  17. Cheeger, J., Naber, A., Valtorta, D.: Critical sets of elliptic equations. Comm. Pure Appl. Math. 68(2), 173–209, 2015

    Article  MathSciNet  Google Scholar 

  18. Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183, 1988

    Article  ADS  MathSciNet  Google Scholar 

  19. Fernández-Real, X., Yu, H.: Generic properties in free boundary problems. arXiv:2308.13209

  20. Fotouhi, M., Shahgholian, H.: A semilinear PDE with free boundary. Nonlinear Anal. 151, 145–163, 2017

    Article  MathSciNet  Google Scholar 

  21. Garofalo, N., Lin, F.-H.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35(2), 245–268, 1986

    Article  MathSciNet  Google Scholar 

  22. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Class. Math. Berlin: Springer, reprint of the 1998 ed. edition, (2001)

  23. Han, Q.: Singular sets of solutions to elliptic equations. Indiana Univ. Math. J. 43(3), 983–1002, 1994

    Article  MathSciNet  Google Scholar 

  24. Han, Q., Hardt, R., Lin, F.-H.: Geometric measure of singular sets of elliptic equations. Comm. Pure Appl. Math. 51(11–12), 1425–1443, 1998

    Article  MathSciNet  Google Scholar 

  25. Knyazik, V.A., Merzhanov, A.G., Solomonov, V.B., Shteinberg, A.S.: Macrokinetics of high-temperature titanium interaction with carbon under electrothermal explosion conditions. Combustion, Explosion and Shock Waves. 21, 1573–8345, 1985

    Article  Google Scholar 

  26. Lin, F.-H.: Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 44(3), 287–308, 1991

    Article  MathSciNet  Google Scholar 

  27. Lindgren, E., Petrosyan, A.: Regularity of the free boundary in a two-phase semilinear problem in two dimensions. Indiana Univ. Math. J. 57(7), 3397–3417, 2008

    Article  MathSciNet  Google Scholar 

  28. Logunov, A.: Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure. Ann. of Math. (2) 187(1), 221–239, 2018

    Article  MathSciNet  Google Scholar 

  29. Monneau, R., Weiss, G.S.: An unstable elliptic free boundary problem arising in solid combustion. Duke Math. J. 136(2), 321–341, 2007

    Article  MathSciNet  Google Scholar 

  30. Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30, 1979

    Article  ADS  MathSciNet  Google Scholar 

  31. Parini, E., Weth, T.: Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems. Math. Z. 280(3–4), 707–732, 2015

    Article  MathSciNet  Google Scholar 

  32. Phillips, D.: Hausdorff measure estimates of a free boundary for a minimum problem. Comm. Partial Diff. Equ. 8(13), 1409–1454, 1983

    Article  MathSciNet  Google Scholar 

  33. Rüland, A.: Unique continuation for sublinear elliptic equations based on Carleman estimates. J. Diff. Equ. 265(11), 6009–6035, 2018

    Article  ADS  MathSciNet  Google Scholar 

  34. Shahgholian, H.: The singular set for the composite membrane problem. Comm. Math. Phys. 271, 93–101, 2007

    Article  ADS  MathSciNet  Google Scholar 

  35. Soave, N., Terracini, S.: The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem. Adv. Math. 334, 243–299, 2018

    Article  MathSciNet  Google Scholar 

  36. Soave, N., Terracini, S.: The nodal set of solutions to some elliptic problems: singular nonlinearities. J. Math. Pures Appl. 9(128), 264–296, 2019

    Article  MathSciNet  Google Scholar 

  37. Soave, N., Weth, T.: The unique continuation property of sublinear equations. SIAM J. Math. Anal. 50(4), 3919–3938, 2018

    Article  MathSciNet  Google Scholar 

  38. Tortone, G.: The nodal set of solutions to some nonlocal sublinear problems. Calc. Var. Partial. Differ. Equ. 61(3), 52, 2022

    Article  MathSciNet  Google Scholar 

  39. Varma, A., Mukasyan, A.S., Hwang, S.: Dynamics of self-propagating reactions in heterogeneous media: experiments and model. Chem. Eng. Sci. 56(4), 1459–1466, 2001

    Article  Google Scholar 

  40. Weiss, G.S.: An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound. 3(2), 121–128, 2001

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are partially supported by the INDAM-GNAMPA group. G. T. is partially supported by the ERC project no. 853404 Variational approach to the regularity of the free boundaries - VAREG held by Bozhidar Velichkov. N. S. is partially supported by the PRIN 2022 project 2022R537CS \(NO^3\) - Nodal Optimization, NOnlinear elliptic equations, NOnlocal geometric problems, with a focus on regularity (European Union - Next Generation EU). Part of this work was carried out while N. S. was visiting the University of Pisa, which he wish to thank for the hospitality. We thank the anonymous referees for the careful reading of the manuscript, and for precious suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Nicola Soave

  2. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Giorgio Tortone

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  1. Nicola Soave
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Correspondence to Nicola Soave.

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Soave, N., Tortone, G. On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity. Arch Rational Mech Anal 248, 26 (2024). https://doi.org/10.1007/s00205-024-01970-4

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