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# A three solution Theorem for Pucci's extremal operator

Dr. Mohan Kumar Mallick, TIFR-CAM
 Speaker Dr. Mohan Kumar Mallick, TIFR-CAM Feb 13, 2020 from 04:00 PM to 05:00 PM LH 006 vCal iCal

Abstract: In this talk we will discuss the existence of three positive solutions to the following boundary value problem:

$$-\mathcal{M}_{\lambda,\Lambda}^+(D^2u)$$  =  $$f(u)$$       $$\rm{in}$$    $$\Omega,\\$$

$$u$$ = $$0$$            $$\rm{on}$$    $$\partial\Omega,$$

where $$\Omega$$ is a bounded smooth domain in $$\mathbb{R}^n$$ and $$f:[0,\infty]\to[0,\infty]$$ is a $$C^{\alpha}$$ function with $$f(0)\geq 0$$. $$\mathcal{M}_{\lambda,\Lambda}^+$$ is the Pucci's maximal operator. The idea of the proof relies on the construction of two pairs of sub-super solutions $$(\psi_1,\phi_1)$$ and $$(\psi_2,\phi_2)$$ where $$\psi_1\leq \psi_2\leq \phi_1$$, $$\psi_1\leq\phi_2\leq \phi_1$$ with $$\psi_2\nless\phi_2$$, and $$\psi_2, \phi_2$$ are strict sub and supersolutions, then we establish the existence of three solutions $$u_1, u_2$$ and $$u_3$$for the above boundary value problem such that $$u_1\in[\psi_1,\phi_2$$, $$u_2\in[\psi_2,\phi_1]$$ and $$u_3\in [\psi_1,\phi_1]\setminus[\psi_1,\phi_2]\cap[\psi_2,\phi_1]$$.

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