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Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

• The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

• The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

• For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

• Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

• We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

• In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

• We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

• Thus the proof could be reduced to a finite number of verified computations: one computation for local minimality and for finding a local minimality neighborhood. A finite number of computations could exhaust the region between the regular $n$-gon and arbitrary $n$-gons verifying the geometric constraints (diameter, inradius, lower bound on smallest side, etc).

Soon we'll also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 8$ using validated computing (interval arithmetics). I'll update the answer once the paper is finished.

Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

• The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

• The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

• For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

• Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

• We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

• In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

• We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

• Thus the proof could be reduced to a finite number of verified computations: one computation for local minimality and for finding a local minimality neighborhood. A finite number of computations could exhaust the region between the regular $n$-gon and arbitrary $n$-gons verifying the geometric constraints (diameter, inradius, lower bound on smallest side, etc).

We also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 6$ using validated computing (interval arithmetics). The arxiv version can be found here

Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

• The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

• The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

• For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

• Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

• We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

• In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

• We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

• Thus the proof could be reduced to a finite number of verified computations.

Soon we'll also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 8$ using validated computing (interval arithmetics). I'll update the answer once the paper is finished.

Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

• The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

• The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

• For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

• Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

• We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

• In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

• We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

• Thus the proof could be reduced to a finite number of verified computations: one computation for local minimality and for finding a local minimality neighborhood. A finite number of computations could exhaust the region between the regular $n$-gon and arbitrary $n$-gons verifying the geometric constraints (diameter, inradius, lower bound on smallest side, etc).

Soon we'll also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 8$ using validated computing (interval arithmetics). I'll update the answer once the paper is finished.

Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality

Soon we'll also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 8$ using validated computing (interval arithmetics). I'll update the answer once the paper is finished.

Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.

Our paper can be found here: On the polygonal Faber-Krahn inequality The key points are:

• The second shape derivative is computed for a simple eigenvalue of a polygon. Classical shape derivative formulas do not apply directly due to the lack of regularity.

• The Hessian matrix of $(x_i,y_i) \mapsto \lambda_1(P)$ is computed explicitly for general $n$-gons

• For regular $n$-gons the eigenvalues of the Hessian matrix for $|P|\lambda_1(P)$ (scale-invariant version) are computed explicitly. They depend on solutions to 3 PDEs.

• Positivity of eigenvalues of the regular $n$-gon is not obvious. We prove that four eigenvalues are $0$ (corresponding to translations, rotations, homotheties which do not change the scale invariant objective). If the remaining $2n-4$ eigenvalues are strictly positive, the local minimality holds.

• We develop explicit numerical estimates for piecewise linear finite elements allowing to guarantee positivity of the remaining $2n-4$ eigenvalues on meshes which are fine enough.

• In the final section we show that the optimal polygon should verify various bounds on geometric elements which can narrow the search space for admissible optimal polygons.

• We also show that the eigenvalues of the Hessian are stable around the regular $n$-gon (fixing two vertices to eliminate the zero eigenvalues). Therefore there exists a quantifiable region of local minimality around the regular $n$-gon (not fully computed).

• Thus the proof could be reduced to a finite number of verified computations.

Soon we'll also have a preprint concerning a complete proof for the local minimality of the regular $n$-gon for $n \leq 8$ using validated computing (interval arithmetics). I'll update the answer once the paper is finished.