On the H\'enon-Lane-Emden conjecture

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $\frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $1<p<\frac{N+2+2a}{N-2}$ (resp., $1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full H\'{e}non-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).Comment: Theorem 4 has been added in the new version. 23 pages, Comments are welcome. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ or http://www.math.ubc.ca/~fazly/research.htm

Gδ-Embeddings in Hilbert space, II

AbstractIn this paper—which is a continuation of [10]—we exhibit some topological conditions on a Banach space which ensure that it contains isometric copies of infinite-dimensional conjugate spaces. This result is used to identify a large class of Banach spaces that are hereditarily separable duals. A method of defining a “Jamestree sum” of a countable number of Banach spaces is given. It is used to construct various counterexamples; for instance, there exists for each integer n a Banach space that can be mapped into Hilbert space via the composition of n but not (n − 1) Gδ-embeddings. We also continue the investigation of the global structure of some geometrically defined Banach spaces. For example, it is shown that a separable Banach space X with the Radon-Nikodym property (R.N.P.) has a subspace y with a boundedly complete finite-dimensional decomposition (F.D.D.) such that XY has an F.D.D. and the R.N.P

Gδ-embeddings in Hilbert space

AbstractIt is shown that a separable Banach space X has the point of weak to norm continuity property (resp. the Radon-Nikodym property) if and only if there exists a compact Gδ-embedding (resp. an Hδ-embedding) from X into l2. This solves several questions of J. Bourgain and H. P. Rosenthal (J. Funct. Anal. 52 (1983)). It is also shown that every non-relatively compact sequence in a Banach space with property (PC) has a difference subsequence which is a boundedly complete basic sequence. This solves a question of Pelczynski and extends some results of W. B. Johnson and H. P. Rosenthal (Studia Math. 43 (1972), 77–92). Various related questions asked in the above Bourgain-Rosenthal reference and by G. A. Edgar and R. F. Wheeler (Pac. J. Math. 115 (1984)) and N. Ghoussoub and H. P. Rosenthal (Math. Ann. 264 (1983), 321–332) are also settled

Vector Valued Subadditive Processes and Applications in Probability

An ergodic theorem is proved which extends the subadditive ergodic theorem of Kingman and the Banach valued ergodic theorem of Mourier. The theorem is applied to several problems, in particular to a problem on empirical distribution functions

The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation \bi u=\f{\lambda}{(1-u)^2}, which models a simple Micro-Electromechanical System (MEMS) device on a ball B\subset\IR^N, under Dirichlet boundary conditions $u=\partial_\nu u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" \la^*>0 such that a stable classical solution u_\la with 0 exists for \la\in (0,\la^*), while there is none of any kind when \la>\la^*. Our main result asserts that the extremal solution $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $N \le 8$ while $u_{\lambda^*}$ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 9$, in which case $1-C_0|x|^{4/3}\leq u_{\lambda^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $C_0:= (\frac{\lambda^*}{\overline{\lambda}})^{1/3}$ and $\bar{\lambda}:= {8/9} (N-{2/3}) (N- {8/3})$.Comment: 19 pages. This paper completes and replaces a paper (with a similar title) which appeared in arXiv:0810.5380. Updated versions --if any-- of this author's papers can be downloaded at this http://www.birs.ca/~nassif

A stationary free boundary problem modeling electrostatic MEMS

A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero

Alloy Design for Additive Manufacturing:Early-Stage Oxidation of Nickel-Based Superalloys

This body of work aims to inform alloy design for additive manufacturing by investigating the early-stage oxidation behavior of Ni-based superalloys processed by laser-powder bed fusion. The oxidation of 14 Ni-based superalloys—some novel and some heritage—at 1000 °C for 24 hours is studied through thermo-gravimetric analysis. The mass gain, oxide layer thickness, oxide scale composition, and depletion γ' zone size are measured. The influence of the alloy composition on these variables is assessed in order to elucidate how increasingly processable and oxidation resistant alloys can be developed. The alloy compositions with Al content greater than 9 at. pct form continuous Al2O3 scales at 1000 °C and display markedly lower parabolic rate constants, mass gain, oxide layer thickness, and γ' depletion zone size. The alloys of lesser Al content have reduced oxidation resistance and formed oxide scales of predominantly Cr2O3. Alloys with Ti content of 2.7 at. pct and greater formed Ti-rich oxide phases in their oxide scales as well as TiN subscale. A trade-off between alloy processability and oxidation resistance is identified, dictated by the deleterious effect of Al content on the ductility dip and the benefit of Al for oxidation resistance. A property space along the pareto front is highlighted which is ideal for having oxidation resistance and processability

Alloy Design for Additive Manufacturing:Early-Stage Oxidation of Nickel-Based Superalloys

This body of work aims to inform alloy design for additive manufacturing by investigating the early-stage oxidation behavior of Ni-based superalloys processed by laser-powder bed fusion. The oxidation of 14 Ni-based superalloys—some novel and some heritage—at 1000 °C for 24 hours is studied through thermo-gravimetric analysis. The mass gain, oxide layer thickness, oxide scale composition, and depletion γ' zone size are measured. The influence of the alloy composition on these variables is assessed in order to elucidate how increasingly processable and oxidation resistant alloys can be developed. The alloy compositions with Al content greater than 9 at. pct form continuous Al2O3 scales at 1000 °C and display markedly lower parabolic rate constants, mass gain, oxide layer thickness, and γ' depletion zone size. The alloys of lesser Al content have reduced oxidation resistance and formed oxide scales of predominantly Cr2O3. Alloys with Ti content of 2.7 at. pct and greater formed Ti-rich oxide phases in their oxide scales as well as TiN subscale. A trade-off between alloy processability and oxidation resistance is identified, dictated by the deleterious effect of Al content on the ductility dip and the benefit of Al for oxidation resistance. A property space along the pareto front is highlighted which is ideal for having oxidation resistance and processability